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

Parameters

Compulsory Input Parameters

1: $n1$ – int64int32nag_int scalar: The number of observations in the first sample, $n_{1}$ .

Constraint: $n1 \geq 1$ .
2: $n2$ – int64int32nag_int scalar: The number of observations in the second sample, $n_{2}$ .

Constraint: $n2 \geq 1$ .
3: $d$ – 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: $0.0 \leq d \leq 1.0$ .

Optional Input Parameters

None.

Output Parameters

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

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

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 nag_stat_prob_kolmogorov2 (g01ez).

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: 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.

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 nag_stat_prob_kolmogorov2 (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 function 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.

Open in the MATLAB editor: g01ez_example

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

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox