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Statistics

  

RousseeuwCrouxQn

  

compute Rousseeuw and Croux' Qn

 

Calling Sequence

Parameters

Description

Computation

Data Set Options

Random Variable Options

Examples

References

Compatibility

Calling Sequence

RousseeuwCrouxQn(A, ds_options)

RousseeuwCrouxQn(X, rv_options)

Parameters

A

-

data set or Matrix data set

X

-

algebraic; random variable or distribution

ds_options

-

(optional) equation(s) of the form option=value where option is one of correction, ignore, or weights; specify options for computing Rousseeuw and Croux' Qn statistic of a data set

rv_options

-

(optional) equation of the form numeric=value; specifies options for computing Rousseeuw and Croux' Qn statistic of a random variable

Description

• 

The RousseeuwCrouxQn function computes a robust measure of the dispersion of the specified data set or random variable, as introduced by Rousseeuw and Croux in [2].

• 

This statistic, referred to as Qn in the remainder of this help page, is defined for a sorted data set A1A2An as:

Qn=OrderStatisticseqseqAiAj,i=j+1..n,j=1..n1,k

  

where k is binomialfloor12n+1,2.

• 

Qn is a robust statistic: it has a high breakdown point (the proportion of arbitrarily large observations it can handle before giving an arbitrarily large result). The breakdown point of Qn is the maximum possible value, 12.

• 

Qn is a measure of dispersion, also called a measure of scale: if Q[n]X=a, then for all real constants α and β, we have Q[n]αX+β=αa.

• 

The first parameter can be a data set, a distribution (see Statistics[Distribution]), a random variable, or an algebraic expression involving random variables (see Statistics[RandomVariable]). For a data set A, RousseeuwCrouxQn computes Qn as defined above. For a distribution or random variable X, RousseeuwCrouxQn computes the asymptotic equivalent - the value that Qn converges to for ever larger samples of X.

Computation

• 

By default, all computations involving random variables are performed symbolically (see option numeric below).

• 

All computations involving data are performed in floating-point; therefore, all data provided must have type/realcons and all returned solutions are floating-point, even if the problem is specified with exact values.

• 

For more information about computation in the Statistics package, see the Statistics[Computation] help page.

Data Set Options

• 

The ds_options argument can contain one or more of the options shown below. More information for some options is available in the Statistics[DescriptiveStatistics] help page.

• 

ignore=truefalse -- This option controls how missing data is handled by the RousseeuwCrouxQn command. Missing items are represented by undefined or Float(undefined). So, if ignore=false and A contains missing data, the RousseeuwCrouxQn command may return undefined. If ignore=true all missing items in A will be ignored. The default value is false.

• 

weights=Vector -- Data weights. The number of elements in the weights array must be equal to the number of elements in the original data sample. By default all elements in A are assigned weight 1.

• 

correction=samplesize or correction=none -- In [2], Rousseeuw and Croux define a correction factor cn for finite sample size as:

dn={0.399n=20.994n=30.512n=40.844n=50.611n=60.857n=70.669n=80.872n=9nn+1.4n>9andnoddnn+3.8n>9andneven

  

If the option correction = samplesize is given, then this correction factor is applied before the result is returned. The default is correction = none, that is, no correction factor is applied.

Random Variable Options

  

The rv_options argument can contain one or more of the options shown below. More information for some options is available in the Statistics[RandomVariables] help page.

• 

numeric=truefalse -- By default, Qn is computed using exact arithmetic. To compute Qn numerically, specify the numeric or numeric = true option.

Examples

withStatistics:

Compute Qn for a data sample.

s1,5,2,2,7,4,1,6,9

s152274169

(1)

RousseeuwCrouxQns

2.

(2)

Employ Rousseeuw and Croux's finite sample size correction.

RousseeuwCrouxQns,'correction=samplesize'

1.74400000000000

(3)

Let's replace four of the values with very large values.

tcopys:

t1..410100:

t

1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000074169

(4)

RousseeuwCrouxQnt

3.

(5)

The value of Qn stays bounded, because it has a high breakdown point.

Compute Qn for a normal distribution.

RousseeuwCrouxQn'Normal'3,5,'numeric'

2.25312055012086

(6)

The symbolic result is an expression involving the inverse (see RootOf) of the error function (see erf). It evaluates to the same floating point number.

RousseeuwCrouxQn'Normal'3,5

10RootOf4erf_Z1

(7)

evalf

2.253120550

(8)

Generate a random sample of size 1000000 from the same distribution and compute the sample's Qn.

ASample'Normal'3,5,1000000:

RousseeuwCrouxQnA

2.25256379473629

(9)

Consider the following Matrix data set.

MMatrix3,1130,114694,4,1527,127368,3,907,88464,2,878,96484,4,995,128007

M31130114694415271273683907884642878964844995128007

(10)

We compute Qn for each of the columns.

RousseeuwCrouxQnM

1.117.12674.

(11)

References

  

[1] Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.

  

[2] Rousseeuw, Peter J., and Croux, Christophe. Alternatives to the Median Absolute Deviation. Journal of the American Statistical Association 88(424), 1993, pp.1273-1283.

Compatibility

• 

The Statistics[RousseeuwCrouxQn] command was introduced in Maple 18.

• 

For more information on Maple 18 changes, see Updates in Maple 18.

See Also

Statistics

Statistics[Computation]

Statistics[DescriptiveStatistics]

Statistics[Distributions]

Statistics[Median]

Statistics[MedianDeviation]

Statistics[RandomVariables]