NAG Library Routine Document
g01ezf (prob_kolmogorov2)
1
Purpose
g01ezf returns the probability associated with the upper tail of the Kolmogorov–Smirnov two sample distribution.
2
Specification
Fortran Interface
Real (Kind=nag_wp)  ::  g01ezf  Integer, Intent (In)  ::  n1, n2  Integer, Intent (Inout)  ::  ifail  Real (Kind=nag_wp), Intent (In)  ::  d 

C Header Interface
#include <nagmk26.h>
double 
g01ezf_ (const Integer *n1, const Integer *n2, const double *d, Integer *ifail) 

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 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: $\mathbf{d}$ – Real (Kind=nag_wp)Input

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: $\mathbf{ifail}$ – IntegerInput/Output

On entry:
ifail must be set to
$0$,
$1\text{or}1$. If you are unfamiliar with this argument you should refer to
Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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
${\mathbf{ifail}}=0$ or
$1$, 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, ${\mathbf{n1}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n2}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n1}}\ge 1$ and ${\mathbf{n2}}\ge 1$.
 ${\mathbf{ifail}}=2$

On entry, ${\mathbf{d}}<0.0$ or ${\mathbf{d}}>1.0$: ${\mathbf{d}}=\u2329\mathit{\text{value}}\u232a$.
 ${\mathbf{ifail}}=3$

The Smirnov approximation used for large samples did not converge in $200$ iterations. The probability is set to $1.0$.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.9 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.8 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
See
Section 3.7 in How to Use the NAG Library and its Documentation for further information.
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
g01ezf is not threaded in any implementation.
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 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 ${n}_{2}$.
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 (g01ezfe.f90)
10.2
Program Data
Program Data (g01ezfe.d)
10.3
Program Results
Program Results (g01ezfe.r)