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# NAG Toolbox: nag_stat_prob_kolmogorov2 (g01ez)

## Purpose

nag_stat_prob_kolmogorov2 (g01ez) returns the probability associated with the upper tail of the Kolmogorov–Smirnov two sample distribution.

## Syntax

[result, ifail] = g01ez(n1, n2, d)
[result, ifail] = nag_stat_prob_kolmogorov2(n1, n2, d)

## 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 nag_stat_prob_kolmogorov2 (g01ez) 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).

## 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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{n1}$int64int32nag_int scalar
The number of observations in the first sample, ${n}_{1}$.
Constraint: ${\mathbf{n1}}\ge 1$.
2:     $\mathrm{n2}$int64int32nag_int scalar
The number of observations in the second sample, ${n}_{2}$.
Constraint: ${\mathbf{n2}}\ge 1$.
3:     $\mathrm{d}$ – double scalar
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: $0.0\le {\mathbf{d}}\le 1.0$.

None.

### Output Parameters

1:     $\mathrm{result}$ – double scalar
The result of the function.
2:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
 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 nag_stat_prob_kolmogorov2 (g01ez).
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## 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 nag_stat_prob_kolmogorov2 (g01ez), 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 function 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 ${n}_{2}$.

## 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.
```function g01ez_example

fprintf('g01ez example results\n\n');

% Upper tail probabilities for 2-sample Kolmogorov–Smirnov distribution.
n1 = [int64( 5);   10;   20;   20;  400;  200; 1000;  200;   15;  100];
n2 = [int64(10);   10;   10;   15;  200;   20;   20;   50;  200;  100];
d  = [0.5;  0.5;  0.5; 0.4833; 0.1412; 0.2861; 0.2113; 0.1796; 0.18; 0.18];

fprintf('     d       n1   n2   two-sided probability\n');
for j = 1:numel(d)

[p, ifail] = g01ez( ...
n1(j), n2(j), d(j));

fprintf('%8.4f%6d%6d%17.4f\n', d(j), n1(j), n2(j), p);
end

```
```g01ez example results

d       n1   n2   two-sided probability
0.5000     5    10           0.3506
0.5000    10    10           0.1678
0.5000    20    10           0.0623
0.4833    20    15           0.0261
0.1412   400   200           0.0083
0.2861   200    20           0.0789
0.2113  1000    20           0.2941
0.1796   200    50           0.1392
0.1800    15   200           0.6926
0.1800   100   100           0.0782
```

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Chapter Introduction
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