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Statistics

  

HodgesLehmann

  

compute Hodges and Lehmann's location estimator

 

Calling Sequence

Parameters

Description

Computation

Data Set Options

Random Variable Options

Examples

References

Compatibility

Calling Sequence

HodgesLehmann(A, ds_options)

HodgesLehmann(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 Hodges and Lehmann's location statistic of a data set

rv_options

-

(optional) equation of the form numeric=value; specifies options for computing Hodges and Lehmann's location statistic of a random variable

Description

• 

The HodgesLehmann function computes a robust measure of the location of the specified data set or random variable, as introduced by Hodges and Lehmann in [2] and independently by Sen in [3]. This statistic is variously called the Hodges-Lehmann-Sen estimator, the Hodges-Lehmann estimator, the Hodges-Lehmann-Sen statistic, or the Hodges-Lehmann statistic.

• 

The Hodges-Lehmann statistic, referred to as HLE in the remainder of this help page, is defined for a data set A1,A2,...,An as:

HLE=12MedianseqseqAi+Aj,i=1..n,j=1..n

• 

The Hodges-Lehmann statistic is a reasonably robust statistic: it has a fairly high breakdown point (the proportion of arbitrarily large observations it can handle before giving an arbitrarily large result). The breakdown point of HLE is 1122 or about 0.29.

• 

The Hodges-Lehmann statistic is a measure of location: if HodgesLehmannX=a, then for all real constants α and β, we have HodgesLehmannαX+β=αa+β.

• 

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, HodgesLehmann computes the Hodges-Lehmann statistic as defined above. For a distribution or random variable X, HodgesLehmann computes the asymptotic equivalent - the value that the Hodges-Lehmann statistic 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 HodgesLehmann command. Missing items are represented by undefined or Float(undefined). So, if ignore=false and A contains missing data, the HodgesLehmann 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.

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, the Hodges-Lehmann statistic is computed using exact arithmetic. To compute the Hodges-Lehmann statistic numerically, specify the numeric or numeric = true option.

Examples

withStatistics:

Compute the Hodges-Lehmann statistic for a data sample.

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

s:=15227416

(1)

HodgesLehmanns

3.50000000000000

(2)

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

tcopys:

t1..210100:

t

1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000227416

(3)

HodgesLehmannt

5.75000000000000

(4)

The Hodges-Lehmann statistic stays bounded, because it has a high breakdown point.

Compute the Hodges-Lehmann statistic for an exponential distribution.

HodgesLehmann'Exponential'1,'numeric'

0.839173495008330

(5)

The symbolic result below evaluates to the same floating point number if the parameter is 1.

HodgesLehmann'Exponential'b

12LambertW1,12ⅇ1+1b

(6)

evalfb=1|b=1

0.8391734950

(7)

Generate a random sample of size 1000000 from the same distribution and compute the sample's Hodges-Lehmann statistic.

ASample'Exponential'1,1000000:

HodgesLehmannA

0.840281350051089

(8)

Consider the following Matrix data set.

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

M:=31130114694415271273683907884642878964844995128007

(9)

We compute the Hodges-Lehmann statistic for each of the columns.

HodgesLehmannM

3.1018.500000000001.11926105

(10)

References

  

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

  

[2] Hodges, Joseph L.; Lehmann, Erich L. Estimation of location based on ranks. Annals of Mathematical Statistics 34 (2), 1963, pp.598-611.

  

[3] Sen, Pranab K. On the estimation of relative potency in dilution(-direct) assays by distribution-free methods. Biometrics 19(4), 1963, pp.532-552.

Compatibility

• 

The Statistics[HodgesLehmann] 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]

 


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