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An analysis of Kolmogorov-Smirnov statistics and their distributions

Authorship This software was created by Melvin Brown Maple Primes: MelvinBrown email: anon-uz39rm6p18ug@base.google.com Acknowledgements The author acknowledges the work of those who created and published the algorithms which this worksheet implements and examines. Such previous work is listed in the References section of this worksheet and referenced at the point of use. Author disclaimer of liability The author makes no representations or warranties, express or implied, nor assumes any liability for the use of this software. Maintenance of notice In the interest of clarity regarding the origin and status of this software, the author requests that any recipient of it maintains these notices affixed to any distribution by the recipient that contains a copy or derivative of this software. 19 July 2011 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

This file: K-Sdistribution.mw MRB: 20/2/2011. Last edit: 19/7/2011. OS and cpu: A) Windows XP, 1.70 GHz single processor, Intel Pentium M. B) Windows 7, 2.4 GHz 4-processor, Intel Core i5.

Overview This worksheet examines the use of Kolomogorov-Smirnov (K-S) statistics and presents of MAPLE implementations of their distributions under different conditions and assumptions. The presentation aims at being representative rather than comprehensive, and so may serve as an introduction to the properties and use of K-S statistics. The worksheet draws on existing work listed in the References section, and it implements transcriptions (by the author of this worksheet) of certain algorithms given in a subset of those references. K-S statistics The two-sided statistic K-S statistic LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJuRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNQ== for samples of n items is described and implemented in [1]. The null hypothesis is that the samples are drawn from a postulated distribution, with the alternative hypothesis that they are not. The one-sided statistics (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) are commonly used, and, sharing a different limiting distribution from that of the two-sided 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 , they represent two different alternative hypotheses to the null; an implementation of the one-sided statistics may be found at [5]. A general introduction to the use of these statistics is given in [4], and a more precise description of their use is given in [8]. A recent implementation of a combination of algorithms for distribution of the two-sided statistic is described in [9]. The group of three statistics is generally referred to as the Kolomogorov-Smirnov (K-S) group of statistics, as used in the K-S group of tests. Preview The worksheet begins with algorithms for the one-sample two-sided Kolmogorov distribution (for the limit of infinite samples) and then for finite samples (Durbin's method [1]), which may be used in K-S statistical tests for whether a sample is drawn from a continuous distribution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==. An important constraint of the method is that if either the form or the parameters of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== are determined from the data LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImlGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn the critical percentile values so determined are invalid. (The probability density function (PDF) and the cumulative distribution function (CDF) are referred to in abbreviate form as the 'density' and 'distribution' respectively.) An example is given of one sample K-S testing. Two sample testing, is then addressed, beginning with a table of approximations for significance levels above 80%. After showing that, in general, the infinite one-sample distribution cannot approximate the two-sample distribution, the exact calculation of the distribution of the two-sample K-S statistic, as described in Kim and Jennrich (1973) [16], is implemented as a MAPLE procedure and its properties illustrated. Finally, the question of handling the possibility of ties in the sample data is addressed. After introducing a MAPLE implementation of Nikiforov's [17] algorithm for testing finite samples in which ties are possible, the effects of ties in tests (inappropriately) using the algorithm of Kim and Jennrich [16] and then, aptly, using the algorithm of Nikiforov's [17] are compared. Finally, to further illustrate the effect of ties, the distributions of the three kinds of K-S statistic are simulated in the case of 200 pairs of randomly drawn samples of uniform density, for sets of sample pairs with ties and without ties.

The infinite sample approximation is easy to implement but gives slightly larger percentile estimates than than the finite sample distribution for a given value of the distance statistic between distributions, and particularly so for small samples sizes.

Regardless of the form of the two-sided Kolmogorov distribution used, the required empirical distribution function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn for n iid observations LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImlGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn is defined as 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

PkkiRkc2ImYqNiRJIm5HRiRJInhHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJComRichIiItSSRTdW1HRiQ2JCZeIyIiIjYjMSZJIlhHRiQ2I0kiaUdGJEYoL0Y5O0YzRidGM0YkRiRGJA==

where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYnLUYsNiUtRi82JVEiWEYnRjJGNS1GIzYlLUYvNiVRImlGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUS8mR3JlYXRlckVxdWFsO0YnL0Y2USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGTy8lKXN0cmV0Y2h5R0ZPLyUqc3ltbWV0cmljR0ZPLyUobGFyZ2VvcEdGTy8lLm1vdmFibGVsaW1pdHNHRk8vJSdhY2NlbnRHRk8vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0Zobi1GLzYlUSJ4RidGMkY1RjJGNUZERks= is the indicator function, equal to 1 if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImlGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn\342\211\244LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and equal to 0 otherwise. The Kolmogorov statistic relative to a reference cumulative distribution function F(x) is: 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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEjRG5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYjNigtRiw2JVEibkYnRi9GMi1GNjYtUSgmIzg1OTQ7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORlgtRiM2KC1GLDYlUSRzdXBGJ0YvRjItRjY2PlEwJkFwcGx5RnVuY3Rpb247RicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnLyUlYm9sZEdGPS9GMEY9LyUqdW5kZXJsaW5lR0Y9LyUqc3Vic2NyaXB0R0Y9LyUsc3VwZXJzY3JpcHRHRj0vJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lJ29wYXF1ZUdGPS8lK2V4ZWN1dGFibGVHRj0vJSlyZWFkb25seUdGPS8lKWNvbXBvc2VkR0Y9LyUqY29udmVydGVkR0Y9LyUraW1zZWxlY3RlZEdGPS8lLHBsYWNlaG9sZGVyR0Y9LyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR0Y9RjlGO0Y+RkBGQkZERkZGSEZXRlktSShtZmVuY2VkR0YkNiQtRiM2Ji1GYnE2KC1GIzYmLUYjNigtRiw2JVEiRkYnRi9GMkZpbi1GYnE2JC1GIzYoRlEtRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSEZXL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRInhGJ0YvRjJGW3BGZXBGOUY5RltwRmVwRjktRjY2LVEoJm1pbnVzO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZgcy1GIzYoRlxyRmluLUZicTYkLUYjNiZGaXJGW3BGZXBGOUY5RltwRmVwRjlGOUY5L0krbXNlbWFudGljc0dGJFEkYWJzRicvJSVvcGVuR1EpJnZlcmJhcjtGJy8lJmNsb3NlR0ZddEZoc0ZbcEZlcEY5RjlGW3BGZXBGOUZbcEZlcEY5RltwRmVwRjk=

The limiting 'large' sample form for the cumulative distribution function of Kolmogorov\342\200\231s statistic LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJuRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSIsRidGNS8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y+LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnRjU= for the 'distance' between a distribution of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= samples and the postulated distribution from which they may be drawn, is 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

LUkmTGltaXRHNiI2JC1JI1ByR0YkNiMxKiYtSSVzcXJ0R0YkNiNJIm5HRiQiIiImSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkRi1GL0kieEdGJC9GLkkpaW5maW5pdHlHRjM=

(Note that the scaling LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVElc3FydEYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJuRicvRjBRJXRydWVGJy9GM1EnaXRhbGljRidGMkYyRjI=, that is applied in the argument here, is not applied to the expression of the finite LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= Kolmogorov distribution that is implemented later in this worK-Sheet. This must be accounted for when comparing these two formulations. The scaling is necessary to for the convergence [2] to the following formulae, the form 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 is sometimes referred to as the standardised statistic for this reason.) The above limit may be expressed as the infinite sum with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZzcmFycjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSlpbmZpbml0eUYnRi9GMkY5 : 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

Zio2JEkieEc2IkkiSkdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYiIiJGKyomIiIjRistSSRzdW1HRiU2JComKSEiIiwmSSJqR0YlRitGK0YzRistSSRleHBHRiU2IywkKihGLUYrKUY1Ri1GKylGJEYtRitGM0YrL0Y1O0YrRiZGK0YzRiVGJUYl

which is the representation originally given by Kolmogorov, or alternatively as: 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

Zio2JEkieEc2IkkiSkdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKigtSSVzcXJ0R0YlNiMsJComIiIjIiIiSSNQaUclKnByb3RlY3RlZEdGMUYxRjFGJCEiIi1JJHN1bUdGJTYkLUkkZXhwR0YlNiMsJCoqI0YxIiIpRjEpLCYqJkYwRjFJImpHRiVGMUYxRjFGNEYwRjEpRjJGMEYxKUYkRjBGNEY0L0ZCO0YxRiZGMUYlRiVGJQ==

which is better suited to small LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RJiZsZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSQwLjNGJ0Y5Rjk= [1]. We refer to these two expressions as respectively the k- and s- expressions. Good approximations of these two expressions can be obtained for finite values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= , but, under practical conditions of finite LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= , differences between expressions (1.4) and (1.5) appear.

We can compare these two expressions of the distribution for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIzEwRidGOUY5 : 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This plot demonstrates difference between the two expression when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and thus the sums in the k- and s-expressions are finite, and it shows that they agree for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RLyZHcmVhdGVyRXF1YWw7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSQwLjNGJ0Y5Rjk=, with the k-expression being grossly in error for small x. The relative difference between these two forms is: 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which confirms that for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RLyZHcmVhdGVyRXF1YWw7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSQwLjNGJ0Y5Rjk= the difference becomes negligible, but it rises very sharply for smaller LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. We can compare the execution time of the k-and s- expressions: 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

LUkjbW5HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JFEwMC4wMDAzOTAwNTAwMDAwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn

LUkjbW5HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JFEwMC4wMDAzNTQwMDAwMDAwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn

So the k-expression is ~20% faster but is limited to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RLyZHcmVhdGVyRXF1YWw7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSMwLkYnRjlGOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEiM0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv. In both forms, the limiting Kolmogorov distribution is obtained when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmc3JhcnI7RidGOUY7Rj5GQEZCRkRGRkZIRkpGTUY1LUYsNiVRKWluZmluaXR5RidGL0YyRjk=. However, note that these are asymptotic expressions for sample size LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZzcmFycjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSlpbmZpbml0eUYnRi9GMkY5. We may need to determine the distribution for finite sample sizes, and that computation is much harder and much researched in the last nearly 80 years. 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 The probability density function (PDF), again in the same large n limit, is: 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

in which we have used the s-expression, as we may wish use the density over the whole range of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RLyZHcmVhdGVyRXF1YWw7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSIwRidGOS1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORldGOQ== and with finite LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= . We can simplify this expression for algorithmic use: 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

The density derived from the k-expression is 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

which is simplified to: 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

We can now examine the approximations to the Kolmogorov distribution for various finite LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. We find that for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIzEwRidGOUY5 the shape of the distribution and its density seems to be independent of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= at least up to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RInxpckYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEjMTBGJ0Y5Rjk=. 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This plot demonstrates the divergence of the finite sums of k- expressions of density and distribution for small LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjxGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRJDAuM0YnRjlGOQ== Hereafter, we work with the s-expression. 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

The first three moments of the density are estimated as: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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEtMC44Njg3MzA0NjQyRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRi8vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JFEtMC44MjI0NjI5NzU0RidGL0YyLUYsNiRRLTAuODQ3NDEzNDkzOUYnRi9GLw==

from which the mean LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbW9HRiQ2LlEkYW5kRicvJSVib2xkR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZFLUYsNi1RIn5GJ0YyRjVGN0Y5RjtGPUY/RkFGQ0ZGLUkjbWlHRiQ2JVEkdGhlRicvJSdpdGFsaWNHRjFGMkZILUZMNiVRKXZhcmlhbmNlRidGT0YyRkhGSC1GTDYlUSRhcmVGJ0ZPRjJGMg== 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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEjbW5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEtMC44Njg3MzA0NjQyRidGOS8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJSlyZWFkb25seUdGPUY5

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEkdmFyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEjOj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRLTAuMDY3NzcwMzU2MEYnRjkvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUpcmVhZG9ubHlHRj1GOQ==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEnJiM5NjM7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIzo9RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkktSSNtbkdGJDYkUS0wLjI2MDMyNzQwMTZGJ0YyLyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lKXJlYWRvbmx5R0YxRjI=

From Section 3 of reference [1], we have: 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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEmbWVhbjFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEtMC44Njg3MzExNjA1RidGOS8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJSlyZWFkb25seUdGPUY5

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEldmFyMUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUS0wLjA2Nzc3MzIwNDRGJ0Y5LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lKXJlYWRvbmx5R0Y9Rjk=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVElc2lnMUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUS0wLjI2MDMzMjg3MjNGJ0Y5LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lKXJlYWRvbmx5R0Y9Rjk=

our approximation with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIzEwRidGOUY5 agrees in its mean and standard deviation with the exact values to 5 significant figures. This is a good verification. The mode is easily found from: 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

LUkjbW5HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JFEtMC43MzU0Njc5MDc5RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn

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Since the usual percentiles for significance testing lie in the domain LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIjxGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkctSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnRjItRiw2JFEiMkYnRi9GLw==, we need to ensure the accuracy of the CDF algorithm particularly in this domain. 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This plot shows that in the area of interest for statistical testing, a value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjRGJ0Y5Rjk= is sufficient, and we know that execution time roughly doubles from J=4 to J=20. To see this another way, we examine the convergence of the distribution as the number of terms LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= in the series increases: 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Our analysis suggests that to obtain sufficient accuracy in the domain [LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR0Y7L0YwUSdpdGFsaWNGJ0Yv], we might set the number of terms 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 i.e. to the lowest integer that exceeds LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIitGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjJGJ0Y5Rjk=. We can include this rule in the algorithm that defines the distribution. 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

We can now compute the percentiles: 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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbW5HRiQ2JFEtMC45NzMwNjMzNzUzRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRi8vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JFEsMS4yMjM4NDc4NzBGJ0YvRjItRiw2JFEsMS4zNTgwOTg2MzlGJ0YvRjItRiw2JFEsMS42Mjc2MjM2MTJGJ0YvRjItRiw2JFEsMS45NDk0NzQ2MDRGJ0YvRi8=

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

In the previous subsection we found the limit J = ceil( x + 2), to be useful as an algorithm for giving estimates of the infinite sample distribution of the scaled K-S statistic x. Thus, 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

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkUHJzR0YlNiRGJC1JJWNlaWxHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywmRiQiIiIiIiNGM0YlRiVGJQ==

We now plot the % difference between the two CDFs - LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjS1NGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkUHJzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw==, the latter with J=20 which we regard as sufficiently close to the true value. 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This plot demonstrates that the chosen algorithm provides sufficient accuracy in the cumulative distribution. We now examine the accuracy of this algorithm for the density function; 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

and plot the difference between the two PDFs LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEja3NGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkcHJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw==, the latter for J=20: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEmcGxvdHNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiYtRiM2JS1GLDYlUShsb2dwbG90RidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRjY2JC1GIzY+LUYsNiVRJGFic0YnL0YwRj9GQC1GNjYkLUYjNigtRiw2JVEja3NGJ0YvRjItRjY2JC1GIzYkLUYsNiVRInhGJ0YvRjJGQEZALUkjbW9HRiQ2LVEoJm1pbnVzO0YnRkAvJSZmZW5jZUdGPy8lKnNlcGFyYXRvckdGPy8lKXN0cmV0Y2h5R0Y/LyUqc3ltbWV0cmljR0Y/LyUobGFyZ2VvcEdGPy8lLm1vdmFibGVsaW1pdHNHRj8vJSdhY2NlbnRHRj8vJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZccC1GLDYlUSRwcnNGJ0YvRjItRjY2JC1GIzYmRmVuLUZpbjYtUSIsRidGQEZcby9GX29GMUZgb0Zib0Zkb0Zmb0Zoby9GW3BRJjAuMGVtRicvRl5wUSwwLjMzMzMzMzNlbUYnLUkjbW5HRiQ2JFEjMjBGJ0ZARkBGQEZARkBGZnAtRmluNi1RIn5GJ0ZARlxvRl5vRmBvRmJvRmRvRmZvRmhvRmpwL0ZecEZbcUZicUZlbi1GaW42LVEiPUYnRkBGXG9GXm9GYG9GYm9GZG9GZm9GaG8vRltwUSwwLjI3Nzc3NzhlbUYnL0ZecEZqcS1GX3E2JFEiMEYnRkAtRmluNi1RIy4uRidGQEZcb0Zeb0Zgb0Zib0Zkb0Zmb0Zob0Zqb0ZlcS1GX3E2JFEiNUYnRkBGZnBGYnEtRiw2JVEmdGl0bGVGJ0YvRjJGZnEtSSNtc0dGJDYjUVdhYnNvbHV0ZX5lcnJvcn5pbn50aGV+YWxnb3JpdGhtfmZvcn5Lb2xtb2dvcm92fmRlbnNpdHlGJ0ZmcC1GLDYlUSdsYWJlbHNGJ0YvRjJGZnEtRjY2Ji1GIzYoLUZpbjY+USFGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlc2l6ZUdRIzEyRicvJSVib2xkR0Y/Rk8vJSp1bmRlcmxpbmVHRj8vJSpzdWJzY3JpcHRHRj8vJSxzdXBlcnNjcmlwdEdGPy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLyUnb3BhcXVlR0Y/Rj0vJSlyZWFkb25seUdGPy8lKWNvbXBvc2VkR0Y/LyUqY29udmVydGVkR0Y/LyUraW1zZWxlY3RlZEdGPy8lLHBsYWNlaG9sZGVyR0Y/LyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR0Y/RkBGXG9GaXBGYG9GYm9GZG9GZm9GaG9GanBGW3ItRmlyNiNGZ25GZnAtRmlyNiNRL2Fic29sdXRlfmVycm9yRidGY3NGQEZARkJGRUZmcEZicS1GLDYlUSVheGVzRidGL0YyRmZxLUYsNiVRJ2ZyYW1lZEYnRi9GMkZmcEZicS1GLDYlUSpncmlkbGluZXNGJ0YvRjJGQEZALUZpbjYtUSI7RidGQEZcb0ZpcEZgb0Zib0Zkb0Zmb0Zob0ZqcEZbckY9RkA=

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This error in the density is sufficiently small for statistical use. This concludes our analysis of the large n Kolmogorov distribution. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2PlEhRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHRjcvJSp1bmRlcmxpbmVHRjcvJSpzdWJzY3JpcHRHRjcvJSxzdXBlcnNjcmlwdEdGNy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLyUnb3BhcXVlR0Y3LyUrZXhlY3V0YWJsZUdGNy8lKXJlYWRvbmx5R0Y3LyUpY29tcG9zZWRHRjcvJSpjb252ZXJ0ZWRHRjcvJStpbXNlbGVjdGVkR0Y3LyUscGxhY2Vob2xkZXJHRjcvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRjcvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjcvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRidGVg==

We now create the s-approximation to the 2-sided infinite sample Kolmogorov distribution in the format of the existing distributions in MAPLE 14. See help: ?Statistics[Distribution]. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRK1N0YXRpc3RpY3NGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIjpGJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVkY9RkA= We use the functions and statistics defined and already calculated above with exception of the mean and variance which are taken from the exact values. 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So now we can apply the MAPLE 14 statistical tools. See the examples in Statistics[Distribution].... LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEsRGVuc2l0eVBsb3RGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2NC1GLDYlUSJaRidGL0YyLUkjbW9HRiQ2LVEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSZyYW5nZUYnRi9GMi1GPjYtUSI9RidGQUZDL0ZHRkVGSEZKRkxGTkZQL0ZTUSwwLjI3Nzc3NzhlbUYnL0ZWRmpuLUkjbW5HRiQ2JFEiMEYnRkEtRj42LVEjLi5GJ0ZBRkNGaG5GSEZKRkxGTkZQL0ZTUSwwLjIyMjIyMjJlbUYnL0ZWRlQtRl1vNiRRIjVGJ0ZBRj0tRiw2JVEqdGhpY2tuZXNzRidGL0YyRmVuLUZdbzYkUSIxRidGQUY9LUYsNiVRJmNvbG9yRidGL0YyRmVuLUYsNiVRJHJlZEYnRi9GMkY9LUYsNiVRKmdyaWRsaW5lc0YnRi9GMkZBRkEtRj42LVEiO0YnRkFGQ0ZGRkhGSkZMRk5GUEZSRltvLyUrZXhlY3V0YWJsZUdGRUZB

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This plot agrees with those plotted earlier. We find that some of the statistical functions in Statistics[Distribution] do not compute for the distribution 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. Are there other methods of computing the distribution? A brief survey of the literature (see for example [1]) shows that the distribution is well-known to be difficult to compute to high precision, and especially so for finite sample sizes.

Accurate evaluation of the finite sample distribution for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJuRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNQ== is hard. See [1]. Here we create a procedure using Durbin's method, as described and implemented by Marsaglia et al. in [1], then test it and compare it with the infinite sample two-sided distribution.

In Durbin's method [1], the distribution is defined by 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 where 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the sample size and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is an LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYvUSNieUYnLyUlYm9sZEdGMS9GM1ElYm9sZEYnLyUrZm9udHdlaWdodEdGVUY7Rj5GQEZCRkRGRkZIRkpGTUY1RitGOQ== matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiSEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjY2LVEifkYnRjlGOy9GP0Y9RkBGQkZERkZGSEZKL0ZORkxGOQ==where 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, whose elements include monomials in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Thus, as reproduced from [1], for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjZGJ0Y5Rjk= the matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is 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The construction of the matrix H is at the core of the Durbin algorithm. *Note that no 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 scaling is applied to the argument 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 of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjUHJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn, in distinction to the infinite sample approximation described in the previous section.

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2PlEhRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHRjcvJSp1bmRlcmxpbmVHRjcvJSpzdWJzY3JpcHRHRjcvJSxzdXBlcnNjcmlwdEdGNy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLyUnb3BhcXVlR0Y3LyUrZXhlY3V0YWJsZUdGNy8lKXJlYWRvbmx5R0Y3LyUpY29tcG9zZWRHRjcvJSpjb252ZXJ0ZWRHRjcvJStpbXNlbGVjdGVkR0Y3LyUscGxhY2Vob2xkZXJHRjcvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRjcvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjcvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRidGVg==

While the MAPLE implementation will not have the speed of the C-code implementation provided in [1], it will enable us to determine the percentile statistics for low values of n. Here we do a step by step implementation, and then verify it against results in [1]. Then, in the next subsection, we implement the steps in a procedure. 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiSEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJStleGVjdXRhYmxlR0Y9Rjk=

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

We now convert H to a MAPLE matrix to use the matrix power capability: 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

LUkjbW5HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JFExMTczMS45MjEzMzg3ODY2MkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw==

So we now have the power of the matrix computed. The required answer is is now: 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

LUkjbW5HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JFEyMC42Mjg0Nzk2MTU0MTg4OTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=

To avoid undefined floats for certain (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiZEYnRi9GMkY5), we use 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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkmbWZyYWNHRiQ2KC1GIzYmLUYjNictSSNtaUdGJDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiIUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR0ZTLyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lKXJlYWRvbmx5R0ZERkBGVkZZRkAtRiM2Ji1JJW1zdXBHRiQ2JUYyRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRlZGWUZALyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Ziby8lKWJldmVsbGVkR0ZELUY9Ni1RIj1GJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjc3Nzc3OGVtRicvRlVGW3AtRiM2KC1JK211bmRlcm92ZXJHRiQ2Jy1GPTYvUSomUHJvZHVjdDtGJy9GV1EuWzE0NCwxNDQsMTQ0XUYnRkAvSSttc2VtYW50aWNzR0YkUSZpbmVydEYnRkJGRS9GSEY4RkkvRkxGOC9GTkY4Rk8vRlJRJjAuMGVtRicvRlVRLDAuMTY2NjY2N2VtRictRiM2KC1GMzYlUSNqbkYnRjZGOUZnby1JI21uR0YkNiRGX29GQEZWRllGQEYyRk8vJSxhY2NlbnR1bmRlckdGRC1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EkNS4wRicvJSZkZXB0aEdGYHIvJSpsaW5lYnJlYWtHUSVhdXRvRictSShtZmVuY2VkR0YkNiQtRiw2KC1GIzYmRmNxRlZGWUZALUYjNiZGMkZWRllGQEZdb0Zgb0Zjb0Zlb0ZARlZGWUZARlZGWUZA

in the computation. 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 Using n=10, d=0.274 from p.2 of [1], where Pr(D[10] \342\211\244 .274) = 0.628479615456504275298526691328 \302\267 \302\267 \302\267, we find agreement to 10 figures. So we now have an algorithm that works and is verified.

Using the insight of the previous subsection, we now implement Durbin's algorithm in a procedure: K:=proc(n::integer,d,ndigits::integer) description "Compute values of the distribution Pr(D[n] <= d) of Kolomogorov's goodness of fit measure D[n] for 1<n<~16000, using the Durbin's method as described and implemented in Marsaglia et al.in Journal of Statistical Software, 8 (18), 1\342\200\2234. (2003). This transcription into MAPLE by Melvin Brown, March 2011: n = number of samples, d = independent variable of the distribution"; local s,k,m,h,H,i,j,jn,mH,Hn,fpn,eH,midm,ef; Digits:=ndigits; #Omit this next line if >7-digit accuracy needed in right tail (but execution times may be high) s:=d*d*n; if(s > 7.2 or (s>3.76 and n>99)) then return evalhf(1.0-2.0*exp(-(2.000071+.331/sqrt(n)+1.409/n)*s)); end if; k := ceil(n*d); m := 2*k-1; h:= k-n*d; if(m < 1)then return print("matrix dimension < 1 at n =",n,"d =",d," in K(n,d,ndigits)"); end if; H := Array(Hessenberg[lower], 1 .. m, 1 .. m, fill = 1); for i from 1 to m do H[i, 1] := H[i, 1]-h^i; H[m, i] := H[m, i]-h^(m-i+1) end do; if (2*h-1 > 0) then H[m, 1] := H[m, 1]+(2*h-1)^m end if; for i from 1 by 1 to m do for j from 1 by 1 to m do if (i-j+1>0) then H[i, j] := H[i, j]/factorial(i-j+1); end if; end do; end do; mH := convert(H, Matrix); Hn := (mH)^n;fpn := 1;for jn to n do fpn := jn*fpn/n end do; evalf(fpn)*Hn[k, k]; end proc: 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

Now run the test against that in p.2 of [1] where they give: Pr(D[10] \342\211\244 .274) = 0.628479615456504275298526 at 20 digit precision. 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

JCI1SHZVXWNhaHolRychIz8=

i.e. we obtain 16 digit agreement, while running LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= with 20 digit execution. In [1], they state: "...we cite values and timings from the C proc below, as well as (20-digit) accuracies provided by a much slower Maple proc. For the C proc, K(2000, .04) = 0.9967694319171325 (.99676943191713676985) takes about 1 second, K(2000, .06) = 0.9999989395692991 (.99999893956930568118) takes 4-5 seconds, but K(16000, .016) = 0.9994523491380971 (0.99945234913828052085 ) takes around 100 seconds, and for n > 4000, getting probabilities such as .999999 can take many minutes." We now report the results for each of these examples using our MAPLE procedure LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5above. comptimeK:=proc(n,dList::list,nC) local d,i,st,xt,s,aK; for d in dList do st:=time(): for i from 1 by 1 to nC do aK:=K(n,d,16); end do; xt:=(evalf(time()-st))/(nC); s:=d*d*n;print(n,d,aK,s,xt); end do; end proc: The following tests were run at 20 digit execution but without using the fit for the right hand tail (i.e with the RH tail fit line #hashed out in the procedure K): 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

NyMkIiIlISIj

NiciJSs/JCIiJSEiIyQiNXFybjg8PlZwbioqISM/JCImKz8kISIlJCIneDZDISIk

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

NyMkIiInISIj

NiciJSs/JCIiJyEiIyQiNTQkbzAkcCZSKikqKioqKiEjPyQiJis/KCEiJSQiJ18zbSEiJA==

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

NyMkIiM7ISIk

NiciJitnIiQiIzshIiQkIjVPTzBHUSJcQlgqKiohIz8kIigrZzQlISInJCIpU2cyNkYm

These three tests give good agreement with [1]. Notice length of time (~3 hrs 5 minutes ,16 digit agreement) taken to compute the last case. This particularly motivates the inclusion of the right hand tail fit to obtain practical execution time for region of probabilities close to 1.0. All three above results give 16-digit agreement with the tests in [1]. This is good validation of our implementation. We now re-run the tests at 16 digit execution, but now using the fit for the right hand tail. This was the lowest precision at which the procedure would run for test 1. 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

NyMkIiIlISIj

NiciJSs/JCIiJSEiIyQiMVZKPD5WcG4qKiEjOyQiJis/JCEiJSQiJzFVPSEiJA==

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

NyMkIiInISIj

NiciJSs/JCIiJyEiIyQiMm95ZCYqWyopKioqKiohIzwkIiYrPyghIiUkIiIhRi4=

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

NyMkIiM7ISIk

NiciJitnIiQiIzshIiQkIjNsLlhRM2FfJSoqKiEjPSQiKCtnNCUhIickIiIhRi4=

Notice the length of time (~3 minutes ,12 digit agreement) taken to compute the first case, in sharp contrast with the latter cases (almost instantaneous, 7 digit agreement) for which the procedure uses the fitted function. This, again, is good validation of our implementation. A general observation from these tests is that we get 4 digits less in precision than the execution precision. These tests are now completed and provide confidence in our implementation of the K(n,d) procedure.

In the large one-sample limit, we expect the equivalence 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 in which LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjUHJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn is the large sample approximation as described above. Notice that the argument of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjUHJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn is the standardised K-S statistic, whereas in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= one uses the raw K-S statistic. We now compare finite sample distributions (at 10 digit execution) with the infinite sample distributions, and we work in the unscaled domain of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= . This means that the argument of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkUHJzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw== must be scaled to bring it into the same domain. (We report results for 50 terms of the s-approximation to the infinite distribution.) 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

Zio2JEkjbnNHNiJJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwmLi1JIktHRiU2JUYkRiYiIzUiIiItSSRQcnNHRiU2JComLUklc3FydEdGJTYjRiRGMEYmRjAiI10hIiJGJUYlRiU=

But first, we first plot LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= for a range of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and compare LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkUHJzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw== for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIzE2RidGOUY5, 64. 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The difference between the infinite and finite sample distributions is clear for n=16,64. We can see that the infinite sample formulation has poor accuracy for the small LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= cases. We now plot the differences for all four sample sizes. 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This plot shows that the finite sample distributions exceed the infinite sample distributions (Prk and Prs - the case shown) in these sample sizes by up to 0.07. At the 95-99 percentiles the difference in cumulative probability is ~ 0.03, but the value of the percentile is likely to have a large error for small n, since the slope of the distribution is quite low at these percentiles. The peak difference occurs at ~ probability of 20%. These results are consistent with Figure 1 in [1]. An alternative way of showing the difference between the infinite and finite sample two-sided Kolmogorov distributions, is to determine the differences between the percentile locations, say for 70%, 95% and 99%. Thus, for finite sample of 10: 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

NiYkIitVJykqMzEkISM1JCIrVW5oJ28kRiUkIitbM1kjNCVGJSQiKyVmOyQqKVtGJQ==

This compares with the percentiles for the infinite sample (using 10 terms in the series summation): 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

NiYkIitzXSVIQSQhIzUkIit6bjlxUUYlJCIrKClcbyVIJUYlJCIrJ3koKnA5JkYl

Thus, 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 .These results are consistent with the finite sample distribution being greater than the finite sample distribution except at the limits [0, infinity). This means that the infinite sample Kolmogorov distribution accepts a larger difference between distributions up to a given percentile level; i.e. it is slightly more tolerant at a given confidence level, so demanding a higher K-S statistic for rejection of the hypothesis.

Here we cross-check our method using the two-sided Kolmogorov distribution, with results given in [5-7]. This illustrates the difference between the one- and two-sided measures and their distributions. 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

IiNd

JCIiJiEiIg==

JCIyOUoiKikqKioqKioqKioqISM8

To make a more comprehensive test we we can determine LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= for various LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSYmbGVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUYsNiVRImRGJ0YvRjJGQUZBRkE= and given LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. 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

NyskIisoSCNlczchIzUkIitbLyRmXiJGJSQiK2AkeVFuIkYlJCIrcltdKCo+RiUkIisySGpvQ0YlJCIrVScpKjMxJEYlJCIrVW5oJ28kRiUkIitbM1kjNCVGJSQiKyVmOyQqKVtGJQ==

and the compare these to the one-sided results given in [5]: 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

NysiIiEkIiRqJCEiJSQiJCZmRiYkIiVWNUYmJCIlOzxGJiQiJSJbI0YmJCIlRUtGJiQiJSdvJEYmJCIlb1hGJg==

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

At this LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIzEwRidGOUY5 sample size we observe the difference between the percentiles of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= for the two-sided and one-sided distributions. For our two-sided 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 we have LUkjbW5HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JFE3MC42Mjg0Nzk2MTU0NTY1MDQyNzUyOUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw== which is in 16 digit agreement with [1], whereas [5] gives LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEnMC44MTM0RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRi8= for the one-sided distribution; this illustrates the difference between these two types of K-S distribution, and the importance of ensuring that the correct distribution is used with the different respective measures of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Tests at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIzUwRidGOUY5 also show a large difference. Taking another source, we find that [6], which is for the two-sided measure, returns LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= as 0.368, 0.410, 0.490 for the 90%, 95% and 99% percentiles, in agreement with our results using LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= which is for the two sided measure. If we compare these three percentiles in [5] with [7], we find that their values of the one-sided measure are in good agreement. The key point is that each measure must be used with its corresponding distribution. There are two one-sided measures, which have the same distribution and therefore provide two results; whereas there is one two-sided measure and its associated distribution.

In this section, we exemplify the use of one sample K-S testing. The distribution of a K-S statistic LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= can be used to obtained confidence levels. At the 95% confidence level LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUkjbW5HRiQ2JFEjOTVGJy9GNlEnbm9ybWFsRictRi82JVEiJUYnRjJGNUYyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+, for example, we expect only 5% of cases to have 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 if the sample data are drawn from the same hypothesised distribution. In this region, we would therefore suspect with odds of at least 19 to 1 that the data were not drawn from the hypothesised distribution. On the otherhand, if we observe a case of 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 we say that with odds greater than 19 to 1 they are from the hypothesised distribution. The key is the choice of hypothesis: a) that samples are or b) are not from the hypothesised distribution. This example illustrate the use of K-S testing, and it also shows how sensitive the test is to the maximum local difference between the empirical and hypothesised distribution.

First, we create an empirical sample of size of 11 drawn from a uniform density i.e. a linear CDF on [0..10]: 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

Ny0jIiNmIiM1IyIjJCpGJSMiI15GJSMiIiQiIiMjIiM5IiImRi8jIiIiRi8jIiNBRi8jIiNMRi8jIiNSRiUiIig=

Ny0jIiIiIiImIyIiJCIiIyMiIzlGJSMiI1IiIzUjIiNBRiVGJSMiI15GLSMiI2ZGLSMiI0xGJSIiKCMiIyQqRi0=

We work with a particular empirical sample: 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

Ny0jIiIiIiImIyIiJCIiIyMiIzlGJSMiI1IiIzUjIiNBRiVGJSMiI15GLSMiI2ZGLSMiI0xGJSIiKCMiIyQqRi0=

The empirical distribution function is constructed [2] as: 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

Zio2JEkibkc2IkkieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiZGJCEiIi1JJHN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkLUkqSGVhdmlzaWRlR0YuNiMsJkYmIiIiJkkiWEdGJTYjSSJqR0YlRisvRjo7RjZGJEY2RiVGJUYl

We wish to compare the hypotheses that this sample was a) drawn from a discrete uniform distribution b) drawn from a continuous uniform distribution. We consider the two hypotheses in turn. To do this we use the two-sided K-S statistic LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJuRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNQ== as given respective in [1] and [2]. 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

For ordered, 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, a purported uniformly distributed set of variates on [0,1], the two-sided Kolmogorov-Smirnov statistic [1] is 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 In passing, we note from [1,5] that the one-sided alternatives proposed by Smirnov - either 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 , the maximum of the \357\254\201rst half of the above list, or 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 , the maximum of the second half - have a common distribution. They are widely used because of Knuth\342\200\231s recommended use of 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 and 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 on the grounds that they \342\200\234seem most convenient for computer use\342\200\235. Here we use the Kolmogorov distribution for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJuRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNQ== , as it provides more accuracy for small LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= . The definition of the two sided K-S statistic LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJuRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNQ== in [1] assumes the hypothesised distribution to be uniform and discrete; it has the form LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkkvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGWC1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYvNiVRInlGJ0YyRjUtRkQ2LVEiLEYnRj5GRy9GS0Y0RkxGTkZQRlJGVC9GV1EmMC4wZW1GJy9GWlEsMC4zMzMzMzMzZW1GJy1GLzYlUSJuRidGMkY1LyUrZXhlY3V0YWJsZUdGSUY+Rj4tRkQ2LVEoJnNyYXJyO0YnRj5GR0ZKRkxGTkZQRlJGVEZhby9GWkZiby1GRDYtUSJ+RidGPkZHRkpGTEZORlBGUkZURmFvRl1wLUYvNiVRKW1heGltaXNlRidGMkY1LUZmbjYkLUYjNiktRi82JVEkc2VxRidGMkY1LUZmbjYkLUYjNi0tRiw2JUZqbi1GIzYkLUYvNiVRImtGJ0YyRjVGPkZALUZENi1RKCZtaW51cztGJ0Y+RkdGSkZMRk5GUEZSRlQvRldRLDAuMjIyMjIyMmVtRicvRlpGanEtSSZtZnJhY0dGJDYoLUYjNiZGY3FGZnFGOkY+LUYjNiRGZW9GPi8lLmxpbmV0aGlja25lc3NHRj0vJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGZ3IvJSliZXZlbGxlZEdGSUZdb0ZjcS1GRDYtUSI9RidGPkZHRkpGTEZORlBGUkZURlZGWUY6LUZENi1RIy4uRidGPkZHRkpGTEZORlBGUkZURmlxRl1wRmVvRmhvRj5GPkZdb0ZocC1GZm42JC1GIzYuLUZdcjYoRmFxRmFyRmNyRmVyRmhyRmpyRmZxRl9xRl5wRl1vRmNxRlxzRjpGX3MtRiw2JUZlby1GRDY+USFGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlc2l6ZUdRIzEyRicvJSVib2xkR0ZJL0YzRkkvJSp1bmRlcmxpbmVHRkkvJSpzdWJzY3JpcHRHRkkvJSxzdXBlcnNjcmlwdEdGSS8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLyUnb3BhcXVlR0ZJRmhvLyUpcmVhZG9ubHlHRkkvJSljb21wb3NlZEdGSS8lKmNvbnZlcnRlZEdGSS8lK2ltc2VsZWN0ZWRHRkkvJSxwbGFjZWhvbGRlckdGSS8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdGSUY+RkdGYG9GTEZORlBGUkZURmFvRllGQEZob0Y+Rj5GaG9GPkY+LUZENi1RIjtGJ0Y+RkdGYG9GTEZORlBGUkZURmFvRllGXnBGaG9GPg==

Zio2JEkieUc2IkkibkdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkpbWF4aW1pc2VHRiU2JC1JJHNlcUclKnByb3RlY3RlZEc2JCwmJkYkNiNJImtHRiUiIiIqJiwmRjRGNUY1ISIiRjVGJkY4RjgvRjQ7RjVGJi1GLjYkLCYqJkY0RjVGJkY4RjVGMkY4RjlGJUYlRiU=

The values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSJ+RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRldGRA==are the values of the empirical function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRIm5GJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInhGJ0YvRjJGQUZBRkE= starting at each of the values 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as previously defined in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Thus, after adding a small value LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5to the sequence of x-values to avoid undefined values at steps, we find the sequence of empirical LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn to be LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEpMC4wMDAwMDFGJ0Y5LUY2Ni1RIjpGJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUYsNiVRJHNlcUYnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYsLUYsNiVRIkZGJ0YvRjItRlo2JC1GIzYpLUZQNiRRIzEwRidGOS1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRInhGJ0YvRjItRjY2LVEiK0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZhcEYrLyUrZXhlY3V0YWJsZUdGPUY5RjlGYm9Gam8tRjY2LVEiPUYnRjlGO0Y+RkBGQkZERkZGSEZKRk0tRlA2JFEiMEYnRjktRjY2LVEjLi5GJ0Y5RjtGPkZARkJGREZGRkhGYHAvRk5GZ29GX29GY3BGOUY5LUY2Ni1RIjtGJ0Y5RjtGZW9GQEZCRkRGRkZIRmZvRk1GY3BGOQ==

Ni0kIiIhRiQkIisrKysrNSEjNSQiKysrKys/RickIisrKysrSUYnJCIrKysrK1NGJyQiKysrKytnRickIisrKysrISlGJyRGJiEiKkYyRjJGMg==

These LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn are measured against the upper and lower values at each step in the hypothesised discrete distribution: 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 The empirical and hypothesised distributions are plotted as follows: 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0Gc(Q*>R***!#9$""*!""7$$"0@nS&*Ra***!#9$""*!""7$$"09y$p*fp***!#9$""*!""7$$"02*o%)*z%)***!#9$""*!""7$$"$+"!""%*undefinedG-%'LEGENDG6#-%)_TYPESETG6#-I#miG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"65Q-hypothesised6"/%'familyGQ0Times~New~Roman6"/%%sizeGQ#126"/%%boldGQ&false6"/%'italicGQ%true6"/%*underlineGQ&false6"/%*subscriptGQ&false6"/%,superscriptGQ&false6"/%+foregroundGQ([0,0,0]6"/%+backgroundGQ.[255,255,255]6"/%'opaqueGQ&false6"/%+executableGQ&false6"/%)readonlyGQ&false6"/%)composedGQ&false6"/%*convertedGQ&false6"/%+imselectedGQ&false6"/%,placeholderGQ&false6"/%6selection-placeholderGQ&false6"/%,mathvariantGQ'italic6"-%&COLORG6&%$RGBG$"#5!""$""!!""$""!!""-%%VIEWG6$;$""!!""$"$5"!"";$""!!""$"#5!""-&%&_AXISG6#"""6'-%+_GRIDLINESG6#%(DEFAULTG-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#"""-%*THICKNESSG6#"""-%-TRANSPARENCYG6#$""!!""-&%&_AXISG6#""#6'-%+_GRIDLINESG6#%(DEFAULTG-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#"""-%*THICKNESSG6#"""-%-TRANSPARENCYG6#$""!!""-%+AXESLABELSG6'Q"x6"Q7cumulative~probability6"-%%FONTG6%%(DEFAULTG%(DEFAULTG"#5%+HORIZONTALG%)VERTICALG-%&TITLEG6$-%)_TYPESETG6#-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"66-I#msG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6#QQEmpirical~and~discrete~hypthesised~distributions6"/%'familyGQ0Times~New~Roman6"/%%sizeGQ#106"/%%boldGQ&false6"/%'italicGQ&false6"/%*underlineGQ&false6"/%*subscriptGQ&false6"/%,superscriptGQ&false6"/%+foregroundGQ([0,0,0]6"/%+backgroundGQ.[255,255,255]6"/%'opaqueGQ&false6"/%+executableGQ&false6"/%)readonlyGQ&false6"/%)composedGQ&false6"/%*convertedGQ&false6"/%+imselectedGQ&false6"/%,placeholderGQ&false6"/%6selection-placeholderGQ&false6"/%0font_style_nameGQ%Text6"/%,mathvariantGQ'normal6"-%-TRANSPARENCYG6#$""!!""-%%ROOTG6'-%)BOUNDS_XG6#$"$5&!""-%)BOUNDS_YG6#$"$q#!""-%-BOUNDS_WIDTHG6#$"%SM!""-%.BOUNDS_HEIGHTG6#$"%]I!""-%)CHILDRENG6"

The deviation of the empirical distribution from the hypothesised distribution is plain to see. We can now compute the statistic LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJuRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNQ== defined as in [1]:: 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

JCIrKyEqKioqKlIhIzU=

which we take to be 0.4. This is larger than the value of 0.3 obtained below using the hypothesis of a continuous distribution, because the hypothesised distribution is discontinuous and and the separation of the empirical distribution from it is measured at the top and the bottom of each of its steps. This value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= can be observed in the above plot at x=7. Inserting this LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY1Rjk=value in the two sided K-S distribution K we get: 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

JCI1UkYnZi5bKnBLeCYqISM/

which is the probability 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 which shows that there is just ~4.3% chance of the value of the K-S statistic exceeding the value 0.4. This is significant at the 5% level and gives quite strong support for rejecting the hypothesis that the samples were drawn from the specified discontinuous distribution.

We now consider the hypothesis that the samples are drawn from a continuous uniform distribution, and use the form of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJuRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNQ== given in [2]. The hypothesised distribution from which the data have been drawn is: 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 in the range LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtbkdGJDYkUSIwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEjMTBGJ0Y0RjRGNC8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y0. We compare the stepwise empirical sample distribution with the hypothesised smooth distribution, as follows: 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The maximum difference between the empirical (black) and the hypothesised (red) distributions can be seen to be 0.3 at x=7. In [2], the one sample two-sided statistic is defined as 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, for which the difference between the empirical and hypothesised distribution is plotted as: 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!#=7$$"0Mn4j`')*\!#9$"0hiE.pjM"!#=7$$"0/3;#z3+]!#9$"0f>Ry?"****!#;7$$"0nLnCi:,&!#9$"0JjE`xV))*!#;7$$"/$f=dOI-&!#8$"0/29GM'p(*!#;7$$"/*zffx)\]!#8$"0,,-/C7]*!#;7$$"/05?'=n2&!#8$"0)\**)z8GB*!#;7$$"06A%4H7$3&!#9$"0*)yd!4xo"*!#;7$$"0sV()>F&*3&!#9$"0"Gc7!GZ5*!#;7$$"0_/MMHF4&!#9$"0xafc1F2*!#;7$$"0Ll!)[Jf4&!#9$"0tY$>^oS!*!#;7$$"0t&RgD`(4&!#9$"0rUgRuY-*!#;7$$"08EFjL"*4&!#9$"0oQFnj'3!*!#;7$$"0`c]qM25&!#9$"0ZV\Hl#**=!#:7$$"0$pQxdL-^!#9$"018EAkw*=!#:7$$"0:IgNW^6&!#9$"0&)pRkb[)=!#:7$$"0PtY$H&z7&!#9$"0jE`1Z?(=!#:7$$"0(Qx/g#3:&!#9$"08E_*R<\=!#:7$$"0Pu[2*pt^!#9$"0jD^#4IE=!#:7$$"0&)pRRX^A&!#9$"0:Igga[x"!#:7$$"0KjEtKpF&!#9$"0oOtEnIs"!#:7$$"0)e<NRZG`!#9$"07C[1E:n"!#:7$$"0>QwAfiP&!#9$"0"=Os2uB;!#:7$$"0v],8QdU&!#9$"0C\)p=Eu:!#:7$$"0b4>G4pZ&!#9$"0X!4=24B:!#:7$$"0oNrK;z_&!#9$"0KkGn$3s9!#:7$$"0d8FM$Q!e&!#9$"0V'Gdmh>9!#:7$$"0'Hf=jfEc!#9$"0.29o.MP"!#:7$$"/'>R=@'yc!#8$"//3;)y8K"!#97$$"0rT$o'f3t&!#9$"0He;LS"p7!#:7$$"0>Qw7,7y&!#9$"0"=Os))z=7!#:7$$"0Rxa*f"p#e!#9$"0hAX+%3t6!#:7$$"08E_p&4ae!#9$"0(Qx/V!f9"!#:7$$"0([(\Rv7)e!#9$"07D]gC(=6!#:7$$"0W)o(eJq)e!#9$"0c6BToH6"!#:7$$"0,-/y(y#*e!#9$"0*zf>A@26!#:7$$"0zen(em&*e!#9$"0@TK7MV5"!#:7$$"0e:J(Ra)*e!#9$"0U%)o-c95"!#:7$$"0(RH@I)***e!#9$"0.1(yp,+6!#:7$$"0Os%p?U,f!#9$"0kF0$zd)4#!#:7$$"0v]w6hG!f!#9$"0D\B))Qr4#!#:7$$"0:He;+V!f!#9$"0&3<M)*p&4#!#:7$$"0Gc7b7e"f!#9$"0sV([u=%3#!#:7$$"0U$oO\KFf!#9$"0e;L1vE2#!#:7$$"0AW)=#3T&f!#9$"0yb6y"*e/#!#:7$$"0.05]"*3)f!#9$"0(\**)\3">?!#:7$$"0nMp)pIGg!#9$"0LlI,$pr>!#:7$$"0v\**GH.3'!#9$"0D]+rq'>>!#:7$$"07BY#o')Hh!#9$"0)oPvJ8q=!#:7$$"08E_/a:='!#9$"0(QxafW==!#:7$$"0sV([!>!Hi!#9$"0Gc7&4)4x"!#:7$$"0oOtY;-G'!#9$"0JjE`$y><!#:7$$"0T#['>'RLj!#9$"0f<N!Qgm;!#:7$$"0=NqI*ozj!#9$"0#['Hp5.i"!#:7$$"0PtY8(oHk!#9$"0jE`'GJq:!#:7$$"0d9HyR8['!#9$"0V&3<-m=:!#:7$$"0b5@#=(=`'!#9$"0X*)y<G"o9!#:7$$"08E_*zJcl!#9$"0(Qx/?oV9!#:7$$"0rT$oTw!e'!#9$"0He;$eB>9!#:7$$"0Kj^**\ve'!#9$"0oO[+]CT"!#:7$$"0$\)>#eL%f'!#9$"02:!yTm09!#:7$$"0L!pyA.'f'!#9$"0n48snRS"!#:7$$"0t&RN(Gxf'!#9$"0F/YErAS"!#:7$$"08,@>D%*f'!#9$"0())*y![d+9!#:7$$"0`1)[;7,m!#9$"0Z$>^$y))R#!#:7$$"0M<Ac9Xg'!#9$"0m#yPa[&R#!#:7$$"09GcZ2zg'!#9$"0'=PCD4#R#!#:7$$"0Or#H"z9i'!#9$"0kG2(3_yB!#:7$$"0d9Hy]]j'!#9$"0U&3<#\\O#!#:7$$"0ze<DS%fm!#9$"/7C[(f0M#!#97$$"0,.1sHQo'!#9$"0)pRz-<;B!#:7$$"0@T#[;"ft'!#9$"0ze<N)3kA!#:7$$"0*yd:c5$y'!#9$"06AWQ%*o@#!#:7$$"0>Pu3,Z$o!#9$"0"Gc7*)Hl@!#:7$$"0kFbS[K)o!#9$"0OsWf^n6#!#:7$$"0.17M%*R$p!#9$"0(Rzec+m?!#:7$$"08D]0,)ep!#9$"0([(\%*)>T?!#:7$$"0AW)oxg$)p!#9$"0yb6B#R;?!#:7$$"0sW9N+,*p!#9$"0Fb&['**)4?!#:7$$"0BXS$Hf'*p!#9$"0xaf12M+#!#:7$$"0N&pzg@)*p!#9$"0l/.#Ry,?!#:7$$"0[X`AR)**p!#9$"0_aYxg,+#!#:7$$"/c*4Pi9+(!#8$"0R/!Hw`)*H!#:7$$"0tXm^&3.q!#9$"0FaL[9p*H!#:7$$"0)f%z!=L1q!#9$"0-a?>oO*H!#:7$$"0BY#*4y&4q!#9$"0x`2!>U!*H!#:7$$"0CZWEjD-(!#9$"0w_btOu(H!#:7$$"0C['H%[b.(!#9$"0w^.d^W'H!#:7$$"0OsW%4cgq!#9$"0kFb0R%RH!#:7$$"0['HfMd&3(!#9$"0_.2aEW"H!#:7$$"0$pQx7tOr!#9$"018EsoK'G!#:7$$"0d9H[lu=(!#9$"0U&3<X`7G!#:7$$"0sV([\3Ms!#9$"0Gc70:fw#!#:7$$"0%ze<h^(G(!#9$"017C)Q[7F!#:7$$"/)f>f0`L(!#8$"/-/3WpkE!#97$$"0)f>R"fiQ(!#9$"0-/3'3u8E!#:7$$"09Fa=G]V(!#9$"0'Gd9=(\c#!#:7$$"0f=P%*z"*[(!#9$"0T"Gc+#3^#!#:7$$"0>PuQrg`(!#9$"0"Gc7'GRY#!#:7$$"0Rxaf$H*e(!#9$"0hAXS12T#!#:7$$"0^,.')*zPw!#9$"0\)pR,?iB!#:7$$"0xa44e3p(!#9$"0AX!4>94B!#:7$$"0Fa3PQmt(!#9$"0tX"H;OjA!#:7$$"0:Heww()y(!#9$"0&3<MKA6A!#:7$$"0/29))R"Ry!#9$"0'Hf=,'3;#!#:7$$"/*zf4q%*)y!#8$"/,-/*H06#!#97$$"0(Qxa^hRz!#9$"08E_%[Qg?!#:7$$"0&*)yd))y()z!#9$"006A96A,#!#:7$$"0mKl!f')R!)!#9$"0LnM4M,'>!#:7$$"0D]+6*\*3)!#9$"0v\**)3]5>!#:7$$"0*zf>dtT")!#9$"0,-/Gk#e=!#:7$$"0v],VA!*=)!#9$"0C\)pv(4"=!#:7$$"0W([(4$GT#)!#9$"0c7D!pre<!#:7$$"0D]+rR8H)!#9$"0v\**Gg'3<!#:7$$"04<M[u7M)!#9$"0"He;bse;!#:7$$"0pQxaJMR)!#9$"0JhAXolg"!#:7$$"/-/3,ZT%)!#8$"/)f>*)H&e:!#97$$"07C[;o1\)!#9$"0)e<N=L4:!#:7$$"0jE`m))\a)!#9$"0PtYL6]X"!#:7$$"0tX"H9<%f)!#9$"0Fa3dGeS"!#:7$$"0%ze<+ZW')!#9$"017C)*HbN"!#:7$$"0;KkQAcp)!#9$"0%yc8wP/8!#:7$$"0oOtOTEu)!#9$"0JjEjetD"!#:7$$"0'>Ry?w#z)!#9$"0/3;#zB27!#:7$$"0oOtE,D%))!#9$"0JjEt)\d6!#:7$$"0Z$pQnq&*))!#9$"0`18E$H/6!#:7$$"0c6BEeE%*)!#9$"0W)oP<Md5!#:7$$"0NpQdIo**)!#9$"0lIhUpJ+"!#:7$$"0rU&3prX!*!#9$"0&Gd94$Ga*!#;7$$"0/29Q(3%4*!#9$"0iHf=E"f!*!#;7$$"0Z%*)y65Y"*!#9$"0Cb5@))*Q&)!#;7$$"0d9HQD$)>*!#9$"0Da3<Yn,)!#;7$$"/%zenTeC*!#8$"0,17C$eTv!#;7$$"0rT$o=I'H*!#9$"0*Ge;8)p.(!#;7$$"06AW3YbM*!#9$"0()yd:RXa'!#;7$$"0'3<Mt#*)R*!#9$"0V"Hems5g!#;7$$"0#Rycp6X%*!#9$"0yg@VI)[b!#;7$$"0%ze<u;)\*!#9$"0e?T#eK=]!#;7$$"0\)pRw.[&*!#9$"0/:IgB'>X!#;7$$"0D\)phU(f*!#9$"0^2:IQd-%!#;7$$"0sW*))R&ok*!#9$"0v_06g9`$!#;7$$"0RycBnsp*!#9$"01;KkFt-$!#;7$$"0KjEV/2v*!#9$"0"oOtc&H\#!#;7$$"0uZ&46q*z*!#9$"/E_/*))H+#!#:7$$"0`067n![)*!#9$"0rW*)yG$>:!#;7$$"0Hd9fp(**)*!#9$"06Fa3/B+"!#;7$$"0D]+^f8&**!#9$"0su\**[S'[!#<7$$"$+"!""$!0;DY-B-6"!#I-%&COLORG6&%$RGBG$""!!""$""!!""$"#5!""-%%VIEWG6$;$""!!""$"$5"!"";$!"&!""$""&!""-&%&_AXISG6#"""6'-%+_GRIDLINESG6#%(DEFAULTG-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#"""-%*THICKNESSG6#"""-%-TRANSPARENCYG6#$""!!""-&%&_AXISG6#""#6'-%+_GRIDLINESG6#%(DEFAULTG-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#"""-%*THICKNESSG6#"""-%-TRANSPARENCYG6#$""!!""-%+AXESLABELSG6'Q%x[j]6"Q.F_n(x)~-~F(x)6"-%%FONTG6%%(DEFAULTG%(DEFAULTG"#5%+HORIZONTALG%)VERTICALG-%&TITLEG6$-%)_TYPESETG6#-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"66-I#msG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"6#Q?F_n(x)~-~F(x)~for~samples~x[j]6"/%'familyGQ0Times~New~Roman6"/%%sizeGQ#106"/%%boldGQ&false6"/%'italicGQ&false6"/%*underlineGQ&false6"/%*subscriptGQ&false6"/%,superscriptGQ&false6"/%+foregroundGQ([0,0,0]6"/%+backgroundGQ.[255,255,255]6"/%'opaqueGQ&false6"/%+executableGQ&false6"/%)readonlyGQ&false6"/%)composedGQ&false6"/%*convertedGQ&false6"/%+imselectedGQ&false6"/%,placeholderGQ&false6"/%6selection-placeholderGQ&false6"/%0font_style_nameGQ%Text6"/%,mathvariantGQ'normal6"-%-TRANSPARENCYG6#$""!!""-%%ROOTG6'-%)BOUNDS_XG6#$"$g&!""-%)BOUNDS_YG6#$"$+#!""-%-BOUNDS_WIDTHG6#$"%!*H!""-%.BOUNDS_HEIGHTG6#$"%SF!""-%)CHILDRENG6"

From this figure, we again see the greatest absolute difference between the samples and the hypothesised distribution is 0.3, and it occurs at x=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiOUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+= 7. More formally, the supremum of the absolute difference is 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

IyIiJCIjNQ==

as observed in the graph above. This is the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= for which we require the probability LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSYmbGVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUYsNiVRImRGJ0YvRjJGQUZBRkE= as given by the two-sided K-S distribution for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIzExRidGOUY5. 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

JCI1dGQoKXkjPVQ0d3UoISM/

This result indicates that we cannot reasonably reject the hypothesis that the sample is drawn from the hypothesised continuous distribution, which we know it is. There is a ~23% chance that a sample with a greater value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= could be drawn from the hypothesised continuous distribution. We therefore favour this continuous over the discrete distribution hypothesis whose statistic was only ~4.3% likely to be exceeded, making it easier to reject.

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

In previous sections, we have compared one sample to a well-defined distribution. The Kolomogorov-Smirnov group of tests may be applied to test whether two one-dimensional probability distributions are drawn from the same distribution. In this case, the empirical two-sided Kolmogorov\342\200\223Smirnov statistic is [2]: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Jy1GLzYlUSJtRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRictSSNtb0dGJDYtUSIsRidGNS8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y+LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYvNiVRIm5GJ0Y9Rj9GPUY/LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRkI2LVEiPUYnRjVGRS9GSEY0RklGS0ZNRk9GUS9GVFEsMC4yNzc3Nzc4ZW1GJy9GV0Zeby1GQjYtUSJ+RidGNUZFRlxvRklGS0ZNRk9GUUZTL0ZXRlUtRiw2JS1GLzYlUSRzdXBGJ0Y9Rj8tRiM2JS1GLzYlUSJ4RidGPUY/Rj1GP0Zmbi1JKG1mZW5jZWRHRiQ2Ji1GIzYqLUYsNiUtRi82JVEiRkYnRj1GPy1GIzYnLUkjbW5HRiQ2JFEiMUYnRjVGQUY6Rj1GP0Zmbi1GX3A2JC1GIzYkRltwRjVGNUZgby1GQjYtUSomdW1pbnVzMDtGJ0Y1RkVGXG9GSUZLRk1GT0ZRL0ZUUSwwLjIyMjIyMjJlbUYnL0ZXRmZxRmBvLUYsNiVGZXAtRiM2Jy1GW3E2JFEiMkYnRjVGQUZZRj1GP0ZmbkZecUY1RjUvJSVvcGVuR1EifGdyRicvJSZjbG9zZUdGYXJGNQ== where 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 with 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 as the indicator function, equal to 1 if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImlGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn\342\211\244LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and equal to 0 otherwise. This may be implemented using the Heaviside function; thus 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 Having determined LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSVtcm93R0YkNiYtRiw2JVEibUYnL0YwUSV0cnVlRicvRjNRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnRjIvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGPC8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSJuRidGO0Y9RjIvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== for a pair of samples, its distribution is required in order determine the statistical significance of its size. In this section, we examine various methods to compute approximate and exact distributions for two sample statistics. See [8] for a clear description of various K-S statistics, and the associated hypotheses. 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

For two samples of finite sizes, the standardised statistic is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkmbXNxcnRHRiQ2Iy1GIzYmLUkmbWZyYWNHRiQ2KC1GIzYnLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RJyZzZG90O0YnL0Y9USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRy8lKXN0cmV0Y2h5R0ZHLyUqc3ltbWV0cmljR0ZHLyUobGFyZ2VvcEdGRy8lLm1vdmFibGVsaW1pdHNHRkcvJSdhY2NlbnRHRkcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZWLUY2NiVRIm5GJ0Y5RjxGOUY8LUYjNidGNS1GQDYtUSIrRidGQ0ZFRkhGSkZMRk5GUEZSL0ZVUSwwLjIyMjIyMjJlbUYnL0ZYRlxvRllGOUY8LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zjby8lKWJldmVsbGVkR0ZHLUZANj5RIUYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJy8lJWJvbGRHRkcvRjpGRy8lKnVuZGVybGluZUdGRy8lKnN1YnNjcmlwdEdGRy8lLHN1cGVyc2NyaXB0R0ZHLyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvJSdvcGFxdWVHRkcvJStleGVjdXRhYmxlR0ZHLyUpcmVhZG9ubHlHRkcvJSljb21wb3NlZEdGRy8lKmNvbnZlcnRlZEdGRy8lK2ltc2VsZWN0ZWRHRkcvJSxwbGFjZWhvbGRlckdGRy8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdGR0ZDRkUvRklGO0ZKRkxGTkZQRlJGVC9GWFEsMC4yNzc3Nzc4ZW1GJ0Y5RjxGPy1JJW1zdWJHRiQ2JS1GNjYlUSJERidGY3BGQy1GIzYlLUY2NiVRI25tRidGOUY8RjlGPC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUZANi1RIn5GJ0ZDRkVGSEZKRkxGTkZQRlJGVEZXLUZANi1RIixGJ0ZDRkVGYHJGSkZMRk5GUEZSRlQvRlhRLDAuMzMzMzMzM2VtRidGYXNGQw==and the position of the curve of the p-value 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 as a function of the standard statistic is approximately independent of the sample sizes. There exist many tables relating the p-value LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEoJmFscGhhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= (or alternatively 1 - 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), the numbers of samples and the K-S statistic. However, many of these tables (see for example [10,15]) provide approximations to the K-S statistics for sample sizes above certain minima for the commonly used levels of significance 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 . Thus, for n, m > 12 in [10] and for n,m > 20 in [15] the approximation to the K-S statistic is defined as 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRKCZhbHBoYTtGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRidGP0Y/LUkjbW9HRiQ2LVEifkYnRj8vJSZmZW5jZUdGPi8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZVRj8=is the standardised K-S statistic for 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 the following table values from [10,15] are combined, since they are consistent at common values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NDU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIi5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSS1GNjYtUSJ+RidGMkY5RjtGPUY/RkFGQ0ZFRkdGSkYy. The 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 and 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 refer to the % p-values for respectively the one- and two-sided K-S statistics. 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEjMTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEiNUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEkMi41RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRi8= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFElMS4yNUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEkMS4wRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRi8= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEkMC41RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRi8= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFElMC4yNUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFElMC4wNUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5NDU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnRjUvRjNRJXRydWVGJy9GNlEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEjMjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEjMTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEiNUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEkMi41RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRi8= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEkMC41RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRi8= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEkMC4xRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRi8= 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFElMS4wN0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFElMS4yMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFElMS4zNkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFElMS40OEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFElMS41MkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFElMS42M0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFElMS43M0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFElMS45N0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv The expression for LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSVtcm93R0YkNiYtRiw2JVEoJmFscGhhO0YnRi9GMi1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEjbm1GJy9GMEZDL0YzUSdpdGFsaWNGJ0YyLyUvc3Vic2NyaXB0c2hpZnRHUSIwRic= at the 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=5% significance level as given in [11] is consistent with the above table. As with all the algorithms presented so far, the above should be used only with no ties, even though it commonly is used otherwise [11]. With ties present, the value of D tends to be too small, and so the p-value is an overestimate. 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

Can the one-sample infinite sample size distribution approximate the two-sample distribution? In general, we cannot use the one-sample procedures for the two-sample case. However, approximations of the two-sample case in the large sample limit might be obtained from the one sample distribution by using the substitution 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 in the asymptotic expression for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjUHJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn. Let us compare the K-S statistics for two commonly used p-values in 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

IiIp

IiIp

NyQkIisoPiRcIXonISM1JCIrZSE9IlEiKUYl

These are the raw K-S statistics, which if standardised become: 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

NyQkIitSJyk0ZTghIiokIis3T2lGO0Yl

and so agree to two decimal places with the values in the table of the previous section. This is also the case for other p-values in that table (for the 2-sided statistics). These results merely explain the basis of tabulated values in the previous section, and the basis of the large n approximations given in the tables of [10, 15]. Can Durbin's one sample method approximate two sample distribution? We now check whether the substitution 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 Durbin's method in procedure K for the single sample testing yields any usable approximation for the two sample testing. Two samples of equal size: Again using common p-values, we have: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEscGVyY2VudGlsZXNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JKG1mZW5jZWRHRiQ2Ji1GIzYmLUkjbW5HRiQ2JFElMC45NUYnRjktRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTlEsMC4zMzMzMzMzZW1GJy1GVTYkUSUwLjk5RidGOUY5RjkvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRjY2LVEiOkYnRjlGO0Y+RkBGQkZERkZGSEZKRk0vJStleGVjdXRhYmxlR0Y9Rjk= 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

NyQkIitAYVFSaSEjNSQiK0dDUVV0RiU=

NyQkIislM3h5QyIhIiokIisnW3clbzlGJQ==

The results, for which the substitution is also integer, only approximate to ~2 decimal places the values given in [15]: Thus we have 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 instead vs [15] with 0.625, 0.75 for sample size 8, and 0.4, 0.5 (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkobWZlbmNlZEdGJDYmLUYjNiktSSNtbkdGJDYkUS0wLjQwOTI0NjA4NDhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGNC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYkUS0wLjQ4ODkzMTY1OTRGJ0Y0LyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUpcmVhZG9ubHlHRkAvJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGJ0Y0RjQvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRjg2LVEifkYnRjRGOy9GP0Y9RkFGQ0ZFRkdGSUZLL0ZPRk0tRjg2LlEkZm9yRicvJSVib2xkR0Y9RjRGO0Zfb0ZBRkNGRUZHRklGS0Zgb0Zcby1JI21pR0YkNiVRJ3NhbXBsZUYnLyUnaXRhbGljR0Y9RjRGXG8tRmdvNiVRJXNpemVGJ0Zqb0Y0RlxvLUYxNiRRJDIwLkYnRjRGNA== Two samples of unequal size In general the substitution 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 in Durbin's method does not work at all if the substitution for n is not integer. The condition for the substitution to be integer is the integer equality: 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is integer and the integers LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJrRidGL0YyRjk= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=. However, this is a very sparse set of (n,m). We can determine the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= that satisfy this condition and then see how close the one sample case LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is to the actual two-sample results. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVElaW1heEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEiOEYnRjktRjY2LVEiO0YnRjlGOy9GP0YxRkBGQkZERkZGSC9GS1EmMC4wZW1GJ0ZNLyUrZXhlY3V0YWJsZUdGPUY5

IiIp

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

NiYiIiNGIyIiIjckJCIrKysrXSgqISM1JCIrKysrXSoqRig=

NiYiIiciIiQiIiM3JCQiK3E2Jyk9JSkhIzUkIis+SypHSCpGKQ==

NiYiIiVGIyIiIzckJCIrcTYnKT0lKSEjNSQiKz5LKkdIKkYo

NiYiIiQiIiciIiM3JCQiK3E2Jyk9JSkhIzUkIis+SypHSCpGKQ==

NiYiIzciIiUiIiQ3JCQiK2lBKWYyKCEjNSQiK2BTLSFIKUYp

NiYiIidGIyIiJDckJCIraUEpZjIoISM1JCIrYFMtIUgpRig=

NiYiIz8iIiYiIiU3JCQiK0BhUVJpISM1JCIrR0NRVXRGKQ==

NiYiIzciIiciIiU3JCQiK0BhUVJpISM1JCIrR0NRVXRGKQ==

NiYiI0kiIiciIiY3JCQiKyUpPnZLYyEjNSQiKzo1SiZvJ0Yp

NiYiI1UiIigiIic3JCQiK1YmPkU+JiEjNSQiK1dEMm1oRik=

This set of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJtRidGNEY3Rj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y+ for which LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= will compute a result is indeed sparse, and we note that the 1st argument of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is indeed LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, the third item in each row. This explains why all results with the same value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= have the same percentile values. Comparing these values to those quoted in [15], we find in the latter that 2,2,1 [no values in [15]] 6,3,2 [5/6, na] 4,4,2 [3/4,na] 3,6,2 [5/6, na] 12,4,3 [2/3,3/4 -- 5/6] 6,6,3 [4/6,5/6] 20,5,4 [na,na] 12,6,4 [7/12,2/3 -- 3/4], which show that the one sample LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= procedure produces values that are broadly consistent with but by no means the same as the values in [15], as well as being limited to a sparse subset set of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJtRidGNEY3Rj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y+. Conclusion With the exception of large sample sizes, for which the substitution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZzcmFycjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1JJm1mcmFjR0YkNigtRiM2Ji1GLDYlUSJtRidGL0YyLUY2Ni1RJyZzZG90O0YnRjlGO0Y+RkBGQkZERkZGSEZKRk1GK0Y5LUYjNidGVC1GNjYtUSIrRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORmpuRitGL0YyLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zhby8lKWJldmVsbGVkR0Y9Rjk= in the infinite sample size distribution gives a good approximation, we conclude that exact two-sample algorithms are required, for the two-sample case.

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

An exact calculation of distribution of the two-sample Kolmogorov statistic is described in Kim and Jennrich (1973) [16] and is listed in Fortran in the Appendix therein. Again, as in all the above sections, it is assumed that the underlying distribution, from which the two samples may be hypothesised to have been drawn, is continuous. This means that, if the method of Kim and Jennrich is to apply, the two samples must not contain any ties. MAPLE implementation The Appendix of [16] includes a listing of the Fortran function AKSCDF which implements the method of Kim and Jennrich. From this listing we create the following MAPLE implementation of the Kim and Jennrich algorithm for the distribution for the two-sample two-sided K-S statistic. Note that the K-S statistic is expressed as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIi9GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xNjY2NjY3ZW1GJy8lJ3JzcGFjZUdGTC1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRIk1GJ0YvRjItRjY2LVEiKkYnRjlGO0Y+L0ZBRj1GQkZERkZGSEZKRk0tRiw2JVEiTkYnRi9GMkY5RjlGOQ== in whose terms the tables in [16] are expressed. To use these tables, one first determines the K-S statistic from the sample distributions, and then determines LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= to look up the value for 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. The procedure simply takes 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 as an input. 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 AKSCDF:=proc(M::integer,N::integer,Dd) description "Computes P(D <=c/(M*N)) using the Kim and Jennrich algorithm [16]. Inputs:M - smaller sample sise; N - larger sample size; Dd = c/(m*n)- K-S criterion. Outputs: value of P(D <=c/(M*N)). This transcription by Melvin Brown, 2011"; local K,i,J,W,u; u:=Array(1..N+1,fill=0); K:=M*N*Dd+0.5;u(1):=1;#print(K); for J from 1 by 1 to N do if(M*J > K) then u(J+1):=0; else u(J+1):=1 end if; end do; for i from 1 by 1 to M do W:=evalf(i/(i+N)); if(N*i > K) then u(1):=0; else u(1):=W*u(1); end if; for J from 1 by 1 to N do if(abs(N*i-M*J) > K) then u(J+1):=0; else u(J+1):=u(J)+u(J+1)*W; end if; end do; end do; u(N+1); end proc: 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 Testing the MAPLE implementation of AKSCDF We now test this procedure to using the tabulated data in [16] and elsewhere. Note that the algorithm computes 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 while the tables given 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. We therefore convert to the latter. 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

IiIp

IiIk

IiNA

JCIrKysrXSgpISM1

JCIqREBAQCIhIzU=

This agrees precisely with the tables in [16] but for the exact location of the step. 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

IiIn

IiIj

IiM1

JCIrTExMTCQpISM1

JCIqNmRHOSghIzU=

This also agrees but for the location of the step. We can also compare with [15]: 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

IiM6

IiM3

JCIrKysrK2IhIzU=

JCIrJm9pJ0gpKiEjNQ==

and find that these results are consistent with [15], when allowance is made for the course grained nature of the distributions We can plot the distribution and compare it to the one-sample case: 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

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLi1JIktHRiU2JSIjP0YkIiM6RiVGJUYl

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This plot shows how the one-sample distribution and the two sample distribution differ. Notably, as n becomes large and m becomes small in the two sample algorithm, the distribution of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiREYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2Jy1GLzYnUSNtbkYnRjIvRjZGNC9GOVEsYm9sZC1pdGFsaWNGJ0Y7RjJGQkZDRjsvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0YyRjhGOw== approaches the distribution of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiREYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2Jy1GLzYnUSJtRidGMi9GNkY0L0Y5USxib2xkLWl0YWxpY0YnRjtGMkZCRkNGOy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjJGOEY7 in the one sample case with m samples. This makes sense, since, with n very large, the distribution of the second sample is an approximation to a continuous distribution. We exemplify this below: 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

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLi1JIktHRiU2JSIjNUYkIiM6RiVGJUYl

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This is a good consistency check on the respective one- and two- sample methods K (Durbin) and AKSCDF (Kim and Jennrich), and their MAPLE implementations. 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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2PlEhRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHRjcvJSp1bmRlcmxpbmVHRjcvJSpzdWJzY3JpcHRHRjcvJSxzdXBlcnNjcmlwdEdGNy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLyUnb3BhcXVlR0Y3LyUrZXhlY3V0YWJsZUdGNy8lKXJlYWRvbmx5R0Y3LyUpY29tcG9zZWRHRjcvJSpjb252ZXJ0ZWRHRjcvJStpbXNlbGVjdGVkR0Y3LyUscGxhY2Vob2xkZXJHRjcvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRjcvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjcvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRidGVg==

The one- and two-sample algorithms, K and AKSCDF, respectively of Marsaglia et al. (Durbin's method) [1] and of Kim and Jennrich [16] are unconditional on ties; they would add 1/n to the probability distribution for each member of the tied group at a given x, but they do not condition the calculation of the underlying distribution of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSNtbkYnL0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnRj1GPy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjU= on the presence of such ties, which really ought to allow for the underlying distributions to be discontinuous - as may be evident in the samples. In effect, these methods assume the samples to be drawn from a continuous underlying distribution; for if this is actually the case, the probability of strict ties is zero. Thus, these methods should only be used with no ties, although they are commonly used otherwise. With ties, the value of D tends to be too small, so that the p-value is an overestimate [11], so a null hypothesis may be accepted when it ought to be rejected. For a more general set of underlying distributions, steps greater than 1/n at a single value of x (abscissum) must be an allowed feature. Nikiforov [17] addresses this more general case, and there has been a recent implementation of that work in the statistical package R [18]. In particular, [17] calculates the underlying distribution of 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 subject to the condition that it accounts for the number and structure of ties in the aggregate of the two sample sets. Thus, if we are not to make the assumption of a continuous underlying distribution, we must adopt the algorithm such as that of [17].

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 We have implemented (as gsmirn) and tested the algorithm GSMIRN of Nikiforov [17] in the MAPLE 15 document Nikiforov.mw, which is available as part of the KSNstat package on the MAPLE Application Centre . This package contains many tests and and examples of the use of gsmirn. Note that the input K-S statistic for gsmirn is not standardised.

To illustrate the effect of ties we compare the results obtained for two samples using Kim & Jennrich's AKSCDF [16] with those obtained using Nikiforov's GSMIRN [17] as implemented in gsmirn. We take a set of pairs of samples and examine the hypothesis that they are drawn from the same underlying distribution. First, we call up an implementation of the library containing gsmirn: 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

NiRRPkM6XFByb2dyYW1+RmlsZXNcTWFwbGV+MTUvbGliNiJRWVM6XE1SQndvcmtcUGhkXFNPRlRXQVJFXE1BUExFXFI0JjVcTVJCbGlicmFyeS9NUkJMaWIubWxhRiQ=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRKEtTTnN0YXRGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIjtGJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRidGPUZA

NyVJJ2dzbWlybkc2IkknZ3N0ZXN0R0YkSSdzdGNhbGNHRiQ=

The two samples which contain ties are: 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

NysiIiIiIiQiIiYiIihGJiIjNkYnIiM5IiM8

NykiIiQiIiYiIzYiIzkiIzoiIzxGKA==

The list of unique categories is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2JVEoY29udmVydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiVRI2x4RidGL0YyLUkjbW9HRiQ2LVEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GPjYtUSJ+RidGQUZDL0ZHRkVGSEZKRkxGTkZQRlIvRlZGVC1GLDYlUSRzZXRGJ0YvRjIvJStleGVjdXRhYmxlR0ZFRkFGQUZYLUY+Ni9RJnVuaW9uRicvJSVib2xkR0YxL0YzUSVib2xkRicvJStmb250d2VpZ2h0R0Zib0ZDRmVuRkhGSkZMRk5GUEZSRmZuRlhGKy1GNjYkLUYjNictRiw2JVEjbHlGJ0YvRjJGPUZnbkZqbkZBRkEtRj42LVEiO0YnRkFGQ0ZGRkhGSkZMRk5GUEZSL0ZWUSwwLjI3Nzc3NzhlbUYnRmpuRkE=

PCoiIiIiIiQiIiYiIigiIzYiIzkiIzoiIzw=

We shall use the K-S statistics to make the comparison of the two methods. Thus, for gsmirn we have 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

IiIq

IiIo

NyoiIiIiIiNGJEYkIiIkRiRGI0Yl

NyUjIiNBIiNqRiMiIiE=

in which dstat[1] is the two-sided statistic, and m is the frequency in each category for the union of the two samples. The p-values for the 3 S statistics are: for kind from 1 by 1 to 3 do pvalueN:=gsmirn(nx,ny,kind,m,dstat[kind]);kind; end do; LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2PlEhRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHRjcvJSp1bmRlcmxpbmVHRjcvJSpzdWJzY3JpcHRHRjcvJSxzdXBlcnNjcmlwdEdGNy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLyUnb3BhcXVlR0Y3LyUrZXhlY3V0YWJsZUdGNy8lKXJlYWRvbmx5R0Y3LyUpY29tcG9zZWRHRjcvJSpjb252ZXJ0ZWRHRjcvJStpbXNlbGVjdGVkR0Y3LyUscGxhY2Vob2xkZXJHRjcvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRjcvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjcvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRidGSEZW

JCIxdGxVdGxVQlYhIzs=

IiIi

JCIxY1MlZjBXND0jISM7

IiIj

IiIi

IiIk

indicating that the hypothesis cannot be rejected at the normal significance levels. We now examine the results obtained using the two-sample algorithm of Kim & Jennrich [16] AKSCDF . We already have the value of the K-S statistics from above: 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

JCIrU0F3KClcISM1

IiIi

JCIrU0F3KClcISM1

IiIj

JCIiIiIiIQ==

IiIk

None of the p-values is statistically significant at the normal levels of testing. The hypothesis that the two samples are from the same distribution cannot therefore be rejected at such levels,whether or not we allow for the presence of ties. But we can still observe the effect of ties. The results show how for kind=1, the p-value given by AKSCDF is greater than that obtained by gsmirn, making it more critical than the latter, and so suggesting less strongly that the null hypothesis cannot be rejected. For kind = 2, the K-S statistic is the same as for kind = 1; AKSCDF gives the same p-value, as might be expected, but gsmirn gives a much lower p-value - even though the K-S statistic is unchanged, suggesting a tendency for first sample values to be smaller than second, but this is no ground for rejecting the null hypothesis. Finally, for kind =3, the K-S statistic is zero; AKSCDF gives p-value = 1, as expected, but gsmirn gives the same p-value of 1.0. For kind =1, 2, which have an identical K-S statistic, the lower value of p-value from gsmirn represents the greater number of ways in which the kind = 2 K-S statistic can be equal or less than that computed from the two samples, if ties in each category are possible; this implies a reduced p-value. The above results give no basis for rejecting the hypothesis that the two samples are drawn from the same distribution, but there is an indication that the first sample distribution tends to be higher than the second i.e. the first sample values tend to be smaller than the second. Inspection of the samples confirms this indication.

Here, directly quoting [8], we remind ourselves of the meaning of three types of statistic: The data consist of two independent samples,one of size n1,denoted by x1 , x2, . . . , xn1,and the other of size n2 denoted by y1, y2, . . . , yn2. Let F (x) and G(x) represent their respective,unknown,distribution functions. Also let S1(x) and S2(x) denote the values of the sample cumulative distribution functions at the point x for the two samples respectively. The Kolmogorov\342\200\223Smirnov statistic tests the null hypothesis H0: F (x) = G(x) against one of the following alternative hypotheses: (i) H1: F (x) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LlErJk5vdEVxdWFsO0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZGRi9GMg==G(x). (ii) H2: F (x) > G(x). This alternative hypothesis is sometimes stated as,\342\200\230The x\342\200\231s tend to be smaller than the y\342\200\231s\342\200\231,i.e.,it would be demonstrated in practical terms if the values of S1(x) tended to exceed the corresponding values of S2(x). (iii) H3: F (x) < G(x). This alternative hypothesis is sometimes stated as,\342\200\230The x\342\200\231s tend to be larger than the y\342\200\231s\342\200\231,i.e.,it would be demonstrated in practical terms if the values of S2(x) tended to exceed the corresponding values of S1(x). One of the following test statistics is computed depending on the particular alternative null hypothesis speci\357\254\201ed (see the description of the parameter KIND). For the alternative hypothesis H1:Dn1,n2 - the largest absolute deviation between the two sample cumulative distribution functions. For the alternative hypothesis H2: D+n1,n2 - the largest positive deviation between the sample cumulative distribution function of the \357\254\201rst sample, S1(x),and the sample cumulative distribution function of the second sample, S2(x). Formally D+n1,n2 = max{S1(x) - S2(x), 0}. For the alternative hypothesis H3:D-n1,n2 - the largest positive deviation between the sample cumulative distribution function of the second sample, S2 (x),and the sample cumulative distribution function of the \357\254\201rst sample, S1(x). Formally D-n1,n2 = max{S2(x) - S1(x), 0}. 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2JVEjbHhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JKG1mZW5jZWRHRiQ2Ji1GIzY1LUkjbW5HRiQ2JFEiMUYnRjktRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTlEsMC4zMzMzMzMzZW1GJy1GVTYkUSIzRidGOUZYLUZVNiRRIjVGJ0Y5RlgtRlU2JFEiN0YnRjlGWC1GVTYkUSI5RidGOUZYLUZVNiRRIzExRidGOUZYLUZVNiRRIzEzRidGOUZYLUZVNiRRIzE1RidGOUZYLUZVNiRRIzE3RidGOS8lK2V4ZWN1dGFibGVHRj1GOUY5LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUY2Ni1RIjtGJ0Y5RjtGZW5GQEZCRkRGRkZIRmZuRk0tRiw2JVEjbHlGJ0YvRjJGNS1GUDYmLUYjNjEtRlU2JFEiMkYnRjlGWC1GVTYkUSI0RidGOUZYLUZVNiRRIzEwRidGOUZYLUZVNiRRIzEyRidGOUZYLUZVNiRRIzE0RidGOUZYLUZVNiRRIzE2RidGOUZYLUZVNiRRIzE4RidGOUZicEY5RjlGZHBGZ3BGanBGYnBGOQ==

NysiIiIiIiQiIiYiIigiIioiIzYiIzgiIzoiIzw=

NykiIiMiIiUiIzUiIzciIzkiIzsiIz0=

The list of unique categories is 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

PDIiIiIiIiMiIiQiIiUiIiYiIigiIioiIzUiIzYiIzciIzgiIzkiIzoiIzsiIzwiIz0=

We shall use the K-S statistics to make the comparison of the two methods. Thus, for gsmirn we have 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

IiIq

IiIo

NzIiIiJGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiM=

NyUjIiM8IiNqRiMjIiIlRiU=

in which dstat[1] is the two-sided statistic, and m is the frequency in each category for the union of the two samples. The p-values for the 3 K-S statistics are: for kind from 1 by 1 to 3 do pvalueN:=gsmirn(nx,ny,kind,m,dstat[kind]);kind; end do; 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

JCIxXyNbPERbPHYpISM7

IiIi

JCIxWF92V192cFshIzs=

IiIj

JCIxPURbPERbPCEqISM7

IiIk

indicating that the hypothesis cannot be rejected at normal significance levels. We now examine the results obtained using the two-sample algorithm of Kim & Jennrich [16] AKSCDF . We already have the value of the two-sided statistic from above: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZILUkjbW9HRiQ2L1EkZm9yRicvJSVib2xkR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y0LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y5LyUpc3RyZXRjaHlHRjkvJSpzeW1tZXRyaWNHRjkvJShsYXJnZW9wR0Y5LyUubW92YWJsZWxpbWl0c0dGOS8lJ2FjY2VudEdGOS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkgtRiw2LVEifkYnL0YzUSdub3JtYWxGJ0Y3RjpGPEY+RkBGQkZERkZGSS1JI21pR0YkNiVRJWtpbmRGJy8lJ2l0YWxpY0dGMS9GM1EnaXRhbGljRidGS0ZLLUYsNi9RJWZyb21GJ0YvRjJGNUY3RjpGPEY+RkBGQkZERkZGSUZLLUkjbW5HRiQ2JFEiMUYnRk5GSy1GLDYvUSNieUYnRi9GMkY1RjdGOkY8Rj5GQEZCRkRGRkZJRktGZW5GSy1GLDYvUSN0b0YnRi9GMkY1RjdGOkY8Rj5GQEZCRkRGRkZJRkstRmZuNiRRIjNGJ0ZORkstRiw2L1EjZG9GJ0YvRjJGNUY3RjpGPEY+RkBGQkZERkZGSUZLLUZRNiVRKXB2YWx1ZUtKRidGVEZWLUYsNi1RKiZjb2xvbmVxO0YnRk5GN0Y6RjxGPkZARkJGRC9GR1EsMC4yNzc3Nzc4ZW1GJy9GSkZccEZlbi1GLDYtUSgmbWludXM7RidGTkY3RjpGPEY+RkBGQkZEL0ZHUSwwLjIyMjIyMjJlbUYnL0ZKRmJwRkstRlE2JVEnQUtTQ0RGRidGVEZWLUkobWZlbmNlZEdGJDYkLUYjNiotRmZuNiRRIjdGJ0ZOLUYsNi1RIixGJ0ZORjcvRjtGMUY8Rj5GQEZCRkRGRi9GSlEsMC4zMzMzMzMzZW1GJy1GZm42JFEiOUYnRk5GX3EtRlE2JVEmZHN0YXRGJ0ZURlYtRmhwNiYtRiM2JUZQLyUrZXhlY3V0YWJsZUdGOUZORk4vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGX3JGTkZOLUYsNi1RIjtGJ0ZORjdGYnFGPEY+RkBGQkZERkZGXXBGS0ZnckZQRmdyRkstRiw2L1EkZW5kRidGL0YyRjVGN0Y6RjxGPkZARkJGREZGRklGS0Zib0ZnckZfckZO

JCIreFBBdyMpISM1

IiIi

JCIreFBBdyMpISM1

IiIj

JCIiIiIiIQ==

IiIk

The results for AKSCDF are unchanged from the tied case, since it does account for them. The two methods give broadly agreeing p-values for kind = 1, AKSCDF suggests that the null hypothesis cannot be rejected at normal levels of significance, but is more critical than gsmirn. For kind = 2, the K-S statistic is the same and so AKSCDF is unchanged, as it does not account for ties; gsmirn is less supportive of the K-S statistic observed, suggesting a tendency for the 1st sample values being lower in value than those of the first samples, but there is no ground for rejecting the null hypothesis. For kind = 3, the K-S statistic is very low, and AKSCDF returns a p-value of 1, suggesting that normally the K-S statistic would certainly be exceeded, so suggesting that the observed statistic is exceptionally low and implying that indeed the first sample values tend to be smaller than the second sample values. On the otherhand, for kind = 3, gsmirn suggests that there is a 10% chance than sample 1 values tend be greater than those of sample 2. Thus, even for kind=3, the null hypothesis cannot be rejected; but this statistic like that of kind=2, suggests the tendency for the 1st sample to have smaller values than sample 2. This outcome can be understood by inspecting the samples in this case. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2PlEhRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHRjcvJSp1bmRlcmxpbmVHRjcvJSpzdWJzY3JpcHRHRjcvJSxzdXBlcnNjcmlwdEdGNy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLyUnb3BhcXVlR0Y3LyUrZXhlY3V0YWJsZUdGNy8lKXJlYWRvbmx5R0Y3LyUpY29tcG9zZWRHRjcvJSpjb252ZXJ0ZWRHRjcvJStpbXNlbGVjdGVkR0Y3LyUscGxhY2Vob2xkZXJHRjcvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRjcvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjcvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRidGSEZW

Here we generate pairs of samples randomly, for the two cases without and with ties. Our aim is to simulate respectively a) the Kim & Jennrich [16] distribution and b) the Nikiforov [17] distribution of K-S statistics. For each case we generate pairs of Ns random samples, whose values lie in the range 0..Mr. 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 We create the sample generating algorithms for case a) (no ties) and case b) (allowing ties) For case a) we sample the real line at high resolution in order to avoid ties: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEjU3pGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRKzEwMDAwMDAwMDBGJ0Y5LUY2Ni1RIjpGJ0Y5RjtGPkZARkJGREZGRkhGSkZNLyUrZXhlY3V0YWJsZUdGPUY5 samNoTies:=proc(Ns,Mr,Sz) description "Generates Ns samples (without ties) from the range of reals 0..Mr; 0..Sz is the range of the random samples before they are ampped in to the range 0..Mr"; local rolla,iir,iter, sam; rolla:=rand(0..Sz); for iter from 1 by 1 to Ns do # iterate to ensure sample without ties is found sam:=seq(evalf(rolla()/Sz*Mr),iir=1..Ns); if (nops({sam}) = Ns) then [sam]; break; elif (iter = Ns) then print("sample without ties not found"); end if; end do; end proc: 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in which there will be no ties, by design of the algorithm above. For case b) we allow ties; and so use the integers rather than the reals: samTies:=proc(Ns,Mr) description "Generates Ns samples (allowing ties) from the range of integers 0..Mr"; local rollb,iir; rollb:=rand(0..Mr); [seq(rollb(),iir=1..Ns)]; end proc: 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The ties are evident in this sample. The above two algorithms can be used to generate pairs of samples for which we can determine the values of the K-S statistics and then determine the distribution of the latter. Notice that the samples are random with a uniform distribution; we may change this to a non-uniform distribution. We now develop the algorithm to compute the K-S statistics between two samples. We make use of the tools in the gsmirn library: 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRKEtTTnN0YXRGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIjtGJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRidGPUZA

NyVJJ2dzbWlybkc2IkknZ3N0ZXN0R0YkSSdzdGNhbGNHRiQ=

We set up parameter values for the sample generation: 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 As examples, first consider a pair of samples with ties allowed: 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

IiM/

IiM/

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NyUjIiIkIiM1I0YkIiM/RiM=

We observe the ties listed in the union distribution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= of the two samples. We compare this with the samples without ties: 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IiM/

N0oiIiJGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiNGI0YjRiM=

NyUjIiIiIiIlRiMjRiQiIzU=

In this case, we observe no ties listed in the union distribution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= of the two samples, as expected. Let us now a create procedure to generate samples and their K-S-statistics and assemble their distributions. We wish a suitable large number of pairs of samples to be tested: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEmbkRpc3RGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjVGJ0Y5LUY2Ni1RIjpGJ0Y5RjtGPkZARkJGREZGRkhGSkZNLyUrZXhlY3V0YWJsZUdGPUY5 distKSstat:=proc(nDist,Ns, Mr, Sz) description "Creates an array of the 3 K-S statistics for nDist pairs of random samples (without and with ties) each comprising Ns samples in the range 0..Mr. For the case of no ties, Sz (e.g. = 100000000) is the large sample size before scaling into the range 0..Mr. The output array has the dimension [ 1(noTies)..2(Ties),1..nDist,[KS1,KS2,KS2]], where KSi is the kind i statistic. Melvin Brown, 2011"; local lx,ly,sstats,id,dstatDist; dstatDist:=Array(1..2,1..nDist,fill=0); for id from 1 by 1 to nDist do lx := samNoTies(Ns, Mr, Sz): ly := samNoTies(Ns, Mr, Sz): sstats := stcalc(lx, ly): dstatDist[1,id] := sstats[4]: lx := samTies(Ns, Mr): ly := samTies(Ns, Mr): sstats := stcalc(lx, ly): dstatDist[2,id] := sstats[4]: end do: dstatDist; end proc: Thus, to create 200 samples of the K-S statistics we have LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYxLUkjbWlHRiQ2JVEmbkRpc3RGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRJDIwMEYnRjktRjY2LVEiOkYnRjlGO0Y+RkBGQkZERkZGSEZKRk0tRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZaLUYsNiVRLHNhbXBLU3N0YXRzRidGL0YyRjUtRiw2JVErZGlzdEtTc3RhdEYnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYtRistRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSEZZL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRI05zRidGL0YyRmFvRlYtRiw2JVEjTXJGJ0YvRjJGYW9GVi1GLDYlUSNTekYnRi9GMi8lK2V4ZWN1dGFibGVHRj1GOUY5LUY2Ni1RIjtGJ0Y5RjtGZG9GQEZCRkRGRkZIRllGTS1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR0ZaLyUmZGVwdGhHRmpwLyUqbGluZWJyZWFrR1EobmV3bGluZUYnLUZmcDYmRmhwRltxRl1xL0ZgcVElYXV0b0YnLUY2Nj5RIUYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJy8lJWJvbGRHRj0vRjBGPS8lKnVuZGVybGluZUdGPS8lKnN1YnNjcmlwdEdGPS8lLHN1cGVyc2NyaXB0R0Y9LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvJSdvcGFxdWVHRj1GYHAvJSlyZWFkb25seUdGPS8lKWNvbXBvc2VkR0Y9LyUqY29udmVydGVkR0Y9LyUraW1zZWxlY3RlZEdGPS8lLHBsYWNlaG9sZGVyR0Y9LyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR0Y9RjlGO0Zkb0ZARkJGREZGRkhGWUZNRmBwRjk=

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

To select the list of samples of the 3 K-S statistics for each of the cases of with and without ties: 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 To plot the distributions of these sampled K-S statistics, we use the definition of the cumulative distribution function at the start of this worksheet. 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This plot shows that for each of the K-S statistics, in the case of samples of equal size 20, the cumulative probability P is higher (p-value is lower), for each K-S statistic, with ties than without ties. Expressed another way, K-S statistics tend to be smaller when ties are present. This is because there are more ways in which a given K-S statistic can be obtained when ties are allowed. Recall, that this result has been obtained for 200 pairs of random samples that are uniformly distributed, and generated using the MAPLE random number generator rand. In particular, instances of a sample pair deviations from the above tendency may be observed. These results simulate the properties of gsmirn (accounts for ties) AKSCDF (does not account for ties). We would therefore expect results from AKSCDF to underestimate P (overestimate p-value), when ties are possible. This means that for samples in which there may be ties AKSCDF is less likely to reject the null hypothesis than it should be i.e. it is less critical of the null hypothesis (that the samples are from the same distribution) than it should be. This applies to all three 3 K-S statistics for the respective alternative hypothesis. 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

[1] Marsaglia, G., Tsang, W. W., Wang, J. (2003) "Evaluating Kolmogorov\342\200\231s Distribution", Journal of Statistical Software, 8 (18), 1\342\200\2234. [2] Kolmogorov\342\200\223Smirnov test, Wikipedia. [3] Engineering Statistics Handbook [4] Short explantory introduction [5] JavaScript implementation of one-sided tests [6] ERI Distance Learning Centre: Table 7 K-S distribution (two-sided) [7] Kolmogorov-Smirnov one-sided test [8] G08CDF \342\200\223 NAG Fortran Library Routine Document [9] Computing the Two-Sided Kolmogorov-Smirnov Distribution, March 2011 [10] Critical Values for the Two-sample Kolmogorov-Smirnov test (2-sided) [11] DEFINITIONS AND FORMULAE WITH STATISTICAL TABLES FOR ELEMENTARY STATISTICS AND QUANTITATIVE METHODS COURSES [12] NAG C Library Function Document: nag_prob_2_sample_ks (g01ezc) [13] Nonparametric Statistics, Bendre, S. M., Hyderbad (Note: the scaling of the statistics appears to be missing the square-root) [14] Journal of Statistical Software [15] Two-Sample Kolmogorov-Smirnov Test [16] Kim, P. J. and Jennrich, R. I. (1973). Tables of the exact sampling distribution of the two-sample Kolmogorov-Smirnov criterion, Dmn, m \342\211\244 n. (Page 79 of Selected Tables in Mathematical Statistics (Edited by H. L. Harter and D. B. Owen), Vol. I. American Mathematical Society, 1973 second print with revisions to original print 1970.) [17] Exact Smirnov two-sample tests for arbitrary distributions, A. Nikiforov, Appl.Stat., vol.43, No. 1. pp.265-270, 1994. (This work rigorously handles the case of ties in the sample data. Follow this link to the author's publication page, where publication, algorithm, FORTRAN and comments can be found. [18] Reference to an implementation of [17] in R can be found here. Thomas Waterhouse, R-developers mailing list, 22 July 2009. We acknowledge reference [2] as the source of the references [1,3-7].

The author would welcome feedback from users on the content of this worksheet, particularly relating to errors that are found or suggestions for improvements.

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