g01ez returns the probability associated with the upper tail of the Kolmogorov–Smirnov two sample distribution.

Syntax

C#
public static double g01ez( int n1, int n2, double d, out int ifail )

Visual Basic
Public Shared Function g01ez ( _ n1 As Integer, _ n2 As Integer, _ d As Double, _ <OutAttribute> ByRef ifail As Integer _ ) As Double

Visual C++
public: static double g01ez( int n1, int n2, double d, [OutAttribute] int% ifail )

F#
static member g01ez : n1 : int * n2 : int * d : float * ifail : int byref -> float

Parameters

n1: Type: System..::..Int32
On entry: the number of observations in the first sample, $n_{1}$ .

Constraint: $n1 \geq 1$ .

n2: Type: System..::..Int32
On entry: the number of observations in the second sample, $n_{2}$ .

Constraint: $n2 \geq 1$ .

d: Type: System..::..Double
On entry: the test statistic $D_{n_{1}, n_{2}}$ , for the two sample Kolmogorov–Smirnov goodness-of-fit test, that is the maximum difference between the empirical cumulative distribution functions (CDFs) of the two samples.

Constraint: $0.0 \leq d \leq 1.0$ .

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Return Value

g01ez returns the probability associated with the upper tail of the Kolmogorov–Smirnov two sample distribution.

Description

Let

F_{n_{1}} (x)

and

G_{n_{2}} (x)

denote the empirical cumulative distribution functions for the two samples, where

n_{1}

and

n_{2}

are the sizes of the first and second samples respectively.

The function g01ez computes the upper tail probability for the Kolmogorov–Smirnov two sample two-sided test statistic

D_{n_{1}, n_{2}}

, where

D_{n_{1}, n_{2}} = \sup_{x} |F_{n_{1}} (x) - G_{n_{2}} (x)| .

The probability is computed exactly if

n_{1}, n_{2} \leq 10000

and

\max (n_{1}, n_{2}) \leq 2500

using a method given by Kim and Jenrich (1973). For the case where

\min (n_{1}, n_{2}) \leq 10 %

of the

\max (n_{1}, n_{2})

and

\min (n_{1}, n_{2}) \leq 80

the Smirnov approximation is used. For all other cases the Kolmogorov approximation is used. These two approximations are discussed in Kim and Jenrich (1973).

References

Conover W J (1980) Practical Nonparametric Statistics Wiley

Feller W (1948) On the Kolmogorov–Smirnov limit theorems for empirical distributions Ann. Math. Statist. 19 179–181

Kendall M G and Stuart A (1973) The Advanced Theory of Statistics (Volume 2) (3rd Edition) Griffin

Kim P J and Jenrich R I (1973) Tables of exact sampling distribution of the two sample Kolmogorov–Smirnov criterion

D_{m n} (m < n)

Selected Tables in Mathematical Statistics 1 80–129 American Mathematical Society

Siegel S (1956) Non-parametric Statistics for the Behavioral Sciences McGraw–Hill

Smirnov N (1948) Table for estimating the goodness of fit of empirical distributions Ann. Math. Statist. 19 279–281

Error Indicators and Warnings

Errors or warnings detected by the method:

$ifail = 1$

On entry,	$n1 < 1$ ,
or	$n2 < 1$ .

$ifail = 2$

On entry,	$d < 0.0$ ,
or	$d > 1.0$ .

$ifail = 3$: The approximation solution did not converge in $500$ iterations. A tail probability of $1.0$ is returned by g01ez.

$ifail = -9000$: An error occured, see message report.

Accuracy

The large sample distributions used as approximations to the exact distribution should have a relative error of less than 5% for most cases.

Parallelism and Performance

None.

Further Comments

The upper tail probability for the one-sided statistics,

D_{n_{1}, n_{2}}^{+}

D_{n_{1}, n_{2}}^{-}

, can be approximated by halving the two-sided upper tail probability returned by g01ez, that is

p / 2

. This approximation to the upper tail probability for either

D_{n_{1}, n_{2}}^{+}

D_{n_{1}, n_{2}}^{-}

is good for small probabilities, (e.g.,

p \leq 0.10

) but becomes poor for larger probabilities.

The time taken by the method increases with

n_{1}

and

n_{2}

, until

n_{1} n_{2} > 10000

\max (n_{1}, n_{2}) \geq 2500

. At this point one of the approximations is used and the time decreases significantly. The time then increases again modestly with

n_{1}

and

n_{2}

Example

The following example reads in

10

different sample sizes and values for the test statistic

D_{n_{1}, n_{2}}

. The upper tail probability is computed and printed for each case.

Example program (C#): g01eze.cs

Example program data: g01eze.d

Example program results: g01eze.r