nag_prob_2_sample_ks (g01ezc) (PDF version)
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g01 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_prob_2_sample_ks (g01ezc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_prob_2_sample_ks (g01ezc) returns the probability associated with the upper tail of the Kolmogorov–Smirnov two sample distribution.

2  Specification

#include <nag.h>
#include <nagg01.h>
double  nag_prob_2_sample_ks (Integer n1, Integer n2, double d, NagError *fail)

3  Description

Let Fn1x and Gn2x denote the empirical cumulative distribution functions for the two samples, where n1 and n2 are the sizes of the first and second samples respectively.
The function nag_prob_2_sample_ks (g01ezc) computes the upper tail probability for the Kolmogorov–Smirnov two sample two-sided test statistic Dn1,n2, where
Dn1,n2=supxFn1x-Gn2x.
The probability is computed exactly if n1,n210000 and maxn1,n22500 using a method given by Kim and Jenrich (1973). For the case where minn1,n2 10 %  of the maxn1,n2 and minn1,n2 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).

4  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 Dmnm<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

5  Arguments

1:     n1IntegerInput
On entry: the number of observations in the first sample, n1.
Constraint: n11.
2:     n2IntegerInput
On entry: the number of observations in the second sample, n2.
Constraint: n21.
3:     ddoubleInput
On entry: the test statistic Dn1,n2, 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: 0.0d1.0.
4:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_CONVERGENCE
The Smirnov approximation used for large samples did not converge in 200 iterations. The probability is set to 1.0.
NE_INT
On entry, n1=value and n2=value.
Constraint: n11 and n21.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL
On entry, d<0.0 or d>1.0: d=value.

7  Accuracy

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

8  Further Comments

The upper tail probability for the one-sided statistics, Dn1,n2+ or Dn1,n2-, can be approximated by halving the two-sided upper tail probability returned by nag_prob_2_sample_ks (g01ezc), that is p/2. This approximation to the upper tail probability for either Dn1,n2+ or Dn1,n2- is good for small probabilities, (e.g., p0.10) but becomes poor for larger probabilities.
The time taken by the function increases with n1 and n2, until n1n2>10000 or maxn1,n22500. At this point one of the approximations is used and the time decreases significantly. The time then increases again modestly with n1 and n2.

9  Example

The following example reads in 10 different sample sizes and values for the test statistic Dn1,n2. The upper tail probability is computed and printed for each case.

9.1  Program Text

Program Text (g01ezce.c)

9.2  Program Data

Program Data (g01ezce.d)

9.3  Program Results

Program Results (g01ezce.r)


nag_prob_2_sample_ks (g01ezc) (PDF version)
g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012