KolmogorovOneSample Class

The class KolmogorovOneSample performs a Kolmogorov-Smirnov goodness-of-fit test in one sample.

Inheritance Hierarchy

SystemObject
Imsl.StatKolmogorovOneSample

Namespace: Imsl.Stat
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

C++

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[SerializableAttribute]
public class KolmogorovOneSample

<SerializableAttribute>
Public Class KolmogorovOneSample

[SerializableAttribute]
public ref class KolmogorovOneSample

[<SerializableAttribute>]
type KolmogorovOneSample =  class end

The KolmogorovOneSample type exposes the following members.

Constructors

	Name	Description
	KolmogorovOneSample	Constructs a one sample Kolmogorov-Smirnov goodness-of-fit test.

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Methods

	Name	Description
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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Properties

	Name	Description
	MaximumDifference	$D^{+}$ , the maximum difference between the theoretical and empirical CDF's.
	MinimumDifference	$D^{-}$ , the minimum difference between the theoretical and empirical CDF's.
	NumberMissing	The number of missing values in the data.
	NumberOfTies	The number of ties in the data.
	OneSidedPValue	Probability of the statistic exceeding D under the null hypothesis of equality and against the one-sided alternative. An exact probability is computed if the number of observation is less than or equal to 80, otherwise an approximate probability is computed.
	TestStatistic	The test statistic, $D = \max(D^{+}, D^{-})$ .
	TwoSidedPValue	Probability of the statistic exceeding D under the null hypothesis of equality and against the two-sided alternative.
	Z	The normalized D statistic without the continuity correction applied.

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Remarks

The hypotheses tested follow:

$\begin{array}{ll} H_0:~ F(x) = F^{*}(x) & H_1:~F(x) \ne F^{*}(x) \\ H_0:~ F(x) \ge F^{*}(x) & H_1:~F(x) \lt F^{*}(x) \\ H_0:~ F(x) \le F^{*}(x) & H_1:~F(x) \gt F^{*}(x) \end{array}$

where

is the cumulative distribution function (CDF) of the random variable, and the theoretical cdf, $F^{*}$ , is specified via the user-supplied function cdf. Let n be the number of observations minus the number of missing observations. The test statistics for both one-sided alternatives $D_n^{+}$ and $D_n^{-}$ and the two-sided

alternative are computed as well as an asymptotic z-score and p-values associated with the one-sided and two-sided hypotheses. For $n \gt 80$ , asymptotic p-values are used (see Gibbons 1971). For $n \le 80$ , exact one-sided p-values are computed according to a method given by Conover (1980, page 350). An approximate two-sided test p-value is obtained as twice the one-sided p-value. The approximation is very close for one-sided p-values less than 0.10 and becomes very bad as the one-sided p-values get larger.

The theoretical CDF is assumed to be continuous. If the CDF is not continuous, the statistics $D_n^{*}$ will not be computed correctly.

Estimation of parameters in the theoretical CDF from the sample data will tend to make the p-values associated with the test statistics too liberal. The empirical CDF will tend to be closer to the theoretical CDF than it should be.

No attempt is made to check that all points in the sample are in the support of the theoretical CDF. If all sample points are not in the support of the CDF, the null hypothesis must be rejected.

Reference

Imsl.Stat Namespace

Other Resources

Example