G01 Chapter Contents
G01 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG01EZF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

G01EZF returns the probability associated with the upper tail of the Kolmogorov–Smirnov two sample distribution, via the routine name.

## 2  Specification

 FUNCTION G01EZF ( N1, N2, D, IFAIL)
 REAL (KIND=nag_wp) G01EZF
 INTEGER N1, N2, IFAIL REAL (KIND=nag_wp) D

## 3  Description

Let ${F}_{{n}_{1}}\left(x\right)$ and ${G}_{{n}_{2}}\left(x\right)$ 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 G01EZF computes the upper tail probability for the Kolmogorov–Smirnov two sample two-sided test statistic ${D}_{{n}_{1},{n}_{2}}$, where
 $Dn1,n2=supxFn1x-Gn2x.$
The probability is computed exactly if ${n}_{1},{n}_{2}\le 10000$ and $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({n}_{1},{n}_{2}\right)\le 2500$ using a method given by Kim and Jenrich (1973). For the case where $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({n}_{1},{n}_{2}\right)\le 10%$ of the $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({n}_{1},{n}_{2}\right)$ and $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({n}_{1},{n}_{2}\right)\le 80$ 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 ${D}_{mn}\left(m 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  Parameters

1:     N1 – INTEGERInput
On entry: the number of observations in the first sample, ${n}_{1}$.
Constraint: ${\mathbf{N1}}\ge 1$.
2:     N2 – INTEGERInput
On entry: the number of observations in the second sample, ${n}_{2}$.
Constraint: ${\mathbf{N2}}\ge 1$.
3:     D – REAL (KIND=nag_wp)Input
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: $0.0\le {\mathbf{D}}\le 1.0$.
4:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
 On entry, ${\mathbf{N1}}<1$, or ${\mathbf{N2}}<1$.
${\mathbf{IFAIL}}=2$
 On entry, ${\mathbf{D}}<0.0$, or ${\mathbf{D}}>1.0$.
${\mathbf{IFAIL}}=3$
The approximation solution did not converge in $500$ iterations. A tail probability of $1.0$ is returned by G01EZF.

## 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.

The upper tail probability for the one-sided statistics, ${D}_{{n}_{1},{n}_{2}}^{+}$ or ${D}_{{n}_{1},{n}_{2}}^{-}$, can be approximated by halving the two-sided upper tail probability returned by G01EZF, that is $p/2$. This approximation to the upper tail probability for either ${D}_{{n}_{1},{n}_{2}}^{+}$ or ${D}_{{n}_{1},{n}_{2}}^{-}$ is good for small probabilities, (e.g., $p\le 0.10$) but becomes poor for larger probabilities.
The time taken by the routine increases with ${n}_{1}$ and ${n}_{2}$, until ${n}_{1}{n}_{2}>10000$ or $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({n}_{1},{n}_{2}\right)\ge 2500$. At this point one of the approximations is used and the time decreases significantly. The time then increases again modestly with ${n}_{1}$ and ${n}_{2}$.

## 9  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.

### 9.1  Program Text

Program Text (g01ezfe.f90)

### 9.2  Program Data

Program Data (g01ezfe.d)

### 9.3  Program Results

Program Results (g01ezfe.r)