NAG Library Function Document
nag_prob_2_sample_ks (g01ezc)
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 ${F}_{{n}_{1}}\left(x\right)$ 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_prob_2_sample_ks (g01ezc) computes the upper tail probability for the Kolmogorov–Smirnov two sample twosided test statistic
${D}_{{n}_{1},{n}_{2}}$, where
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 ${D}_{mn}\left(m<n\right)$ Selected Tables in Mathematical Statistics 1 80–129 American Mathematical Society
Siegel S (1956) Nonparametric 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:
$\mathbf{n1}$ – IntegerInput

On entry: the number of observations in the first sample, ${n}_{1}$.
Constraint:
${\mathbf{n1}}\ge 1$.
 2:
$\mathbf{n2}$ – IntegerInput

On entry: the number of observations in the second sample, ${n}_{2}$.
Constraint:
${\mathbf{n2}}\ge 1$.
 3:
$\mathbf{d}$ – doubleInput

On entry: the test statistic ${D}_{{n}_{1},{n}_{2}}$, for the two sample Kolmogorov–Smirnov goodnessoffit 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:
$\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see
Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.2.1.2 in the Essential Introduction for further information.
 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, ${\mathbf{n1}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n2}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n1}}\ge 1$ and ${\mathbf{n2}}\ge 1$.
 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.
An unexpected error has been triggered by this function. Please contact
NAG.
See
Section 3.6.6 in the Essential Introduction for further information.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 3.6.5 in the Essential Introduction for further information.
 NE_REAL

On entry, ${\mathbf{d}}<0.0$ or ${\mathbf{d}}>1.0$: ${\mathbf{d}}=\u2329\mathit{\text{value}}\u232a$.
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 Parallelism and Performance
Not applicable.
The upper tail probability for the onesided statistics, ${D}_{{n}_{1},{n}_{2}}^{+}$ or ${D}_{{n}_{1},{n}_{2}}^{}$, can be approximated by halving the twosided upper tail probability returned by nag_prob_2_sample_ks (g01ezc), 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 ${n}_{2}$.
10 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.
10.1 Program Text
Program Text (g01ezce.c)
10.2 Program Data
Program Data (g01ezce.d)
10.3 Program Results
Program Results (g01ezce.r)