NAG Library Routine Document
g01eyf (prob_kolmogorov1)
1
Purpose
g01eyf returns the upper tail probability associated with the one sample Kolmogorov–Smirnov distribution, via the routine name.
2
Specification
Fortran Interface
Real (Kind=nag_wp)  ::  g01eyf  Integer, Intent (In)  ::  n  Integer, Intent (Inout)  ::  ifail  Real (Kind=nag_wp), Intent (In)  ::  d 

C Header Interface
#include nagmk26.h
double 
g01eyf_ (const Integer *n, const double *d, Integer *ifail) 

3
Description
Let ${S}_{n}\left(x\right)$ be the sample cumulative distribution function and ${F}_{0}\left(x\right)$ the hypothesised theoretical distribution function.
g01eyf returns the upper tail probability,
$p$, associated with the onesided Kolmogorov–Smirnov test statistic
${D}_{n}^{+}$ or
${D}_{n}^{}$, where these onesided statistics are defined as follows;
If
$n\le 100$ an exact method is used; for the details see
Conover (1980). Otherwise a large sample approximation derived by Smirnov is used; see
Feller (1948),
Kendall and Stuart (1973) or
Smirnov (1948).
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
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: $\mathbf{n}$ – IntegerInput

On entry: $n$, the number of observations in the sample.
Constraint:
${\mathbf{n}}\ge 1$.
 2: $\mathbf{d}$ – Real (Kind=nag_wp)Input

On entry: contains the test statistic, ${D}_{n}^{+}$ or ${D}_{n}^{}$.
Constraint:
$0.0\le {\mathbf{d}}\le 1.0$.
 3: $\mathbf{ifail}$ – 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, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 1$.
 ${\mathbf{ifail}}=2$

On entry, ${\mathbf{d}}<0.0$ or ${\mathbf{d}}>1.0$: ${\mathbf{d}}=\u2329\mathit{\text{value}}\u232a$.
 ${\mathbf{ifail}}=99$
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 distribution used as an approximation to the exact distribution should have a relative error of less than $2.5$% for most cases.
8
Parallelism and Performance
g01eyf is not threaded in any implementation.
The upper tail probability for the twosided statistic, ${D}_{n}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({D}_{n}^{+},{D}_{n}^{}\right)$, can be approximated by twice the probability returned via g01eyf, that is $2p$. (Note that if the probability from g01eyf is greater than $0.5$ then the twosided probability should be truncated to $1.0$). This approximation to the tail probability for ${D}_{n}$ is good for small probabilities, (e.g., $p\le 0.10$) but becomes very poor for larger probabilities.
The time taken by the routine increases with $n$, until $n>100$. At this point the approximation is used and the time decreases significantly. The time then increases again modestly with $n$.
10
Example
The following example reads in $10$ different sample sizes and values for the test statistic ${D}_{n}$. The upper tail probability is computed and printed for each case.
10.1
Program Text
Program Text (g01eyfe.f90)
10.2
Program Data
Program Data (g01eyfe.d)
10.3
Program Results
Program Results (g01eyfe.r)