KolmogorovTwoSample Class

Performs a Kolmogorov-Smirnov two-sample test.

Inheritance Hierarchy

System.Object
Imsl.Stat.KolmogorovTwoSample

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

Syntax

C++

Copy

[SerializableAttribute]
public class KolmogorovTwoSample

<SerializableAttribute>
Public Class KolmogorovTwoSample

[SerializableAttribute]
public ref class KolmogorovTwoSample

[<SerializableAttribute>]
type KolmogorovTwoSample =  class end

The KolmogorovTwoSample type exposes the following members.

Constructors

	Name	Description
	KolmogorovTwoSample	Constructs a two 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.
	NumberMissingX	Returns the number of missing values in the x sample.
	NumberMissingY	The number of missing values in the y sample.
	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. This probability is twice the probability, , reported by OneSidedPValue, (or 1.0 if $p_1 \ge 1/2$ ). This approximation is nearly exact when $p_1 \lt 0.1$ .
	Z	The normalized D statistic without the continuity correction applied.

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Remarks

Class KolmogorovTwoSample computes Kolmogorov-Smirnov two-sample test statistics for testing that two continuous cumulative distribution functions (CDF's) are identical based upon two random samples. One- or two-sided alternatives are allowed. Exact p-values are computed for the two-sided test when $nm \le 104$ , where n is the number of non-missing X observations and m the number of non-missing Y observation.

Let denote the empirical CDF in the X sample, let denote the empirical CDF in the Y sample and let the corresponding population distribution functions be denoted by and , respectively. Then, the hypotheses tested by KolmogorovTwoSample are as follows:

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

The test statistics are given as follows:

$\begin{array}{rl} D_{mn} & = \max(D_{mn}^{+}, D_{mn}^{-}) \\ D_{mn}^{+} & = \max_x(F_n(x)-G_m(x)) \\ D_{mn}^{-} & = \max_x(G_m(x)-F_n(x)) \end{array}$

Asymptotically, the distribution of the statistic

$Z = D_{mn} \sqrt{\frac{m+n}{mn}}$

converges to a distribution given by Smirnov (1939).

Exact probabilities for the two-sided test are computed when $nm \le 104$ , according to an algorithm given by Kim and Jennrich (1973). When $nm \gt 104$ , the very good approximations given by Kim and Jennrich are used to obtain the two-sided p-values. The one-sided probability is taken as one half the two-sided probability. This is a very good approximation when the p-value is small (say, less than 0.10) and not very good for large p-values.

Reference

Imsl.Stat Namespace

Other Resources

Example