Syntax

C#
[SerializableAttribute] public class KolmogorovTwoSample

Visual Basic (Declaration)
<SerializableAttribute> _ Public Class KolmogorovTwoSample

Visual C++
[SerializableAttribute] public ref class KolmogorovTwoSample

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.

Inheritance Hierarchy

System..::.Object
Imsl.Stat..::.KolmogorovTwoSample

Syntax

Remarks

Inheritance Hierarchy

See Also