NAG Library Function Document
nag_prob_1_sample_ks (g01eyc)
1 Purpose
nag_prob_1_sample_ks (g01eyc) returns the upper tail probability associated with the one sample Kolmogorov–Smirnov distribution.
2 Specification
#include <nag.h> 
#include <nagg01.h> 
double 
nag_prob_1_sample_ks (Integer n,
double d,
NagError *fail) 

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.
nag_prob_1_sample_ks (g01eyc) 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:
$\mathbf{d}$ – doubleInput

On entry: contains the test statistic, ${D}_{n}^{+}$ or ${D}_{n}^{}$.
Constraint:
$0.0\le {\mathbf{d}}\le 1.0$.
 3:
$\mathbf{fail}$ – 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_INT

On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 1$.
 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 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
Not applicable.
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 nag_prob_1_sample_ks (g01eyc), that is $2p$. (Note that if the probability from nag_prob_1_sample_ks (g01eyc) 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 function 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 (g01eyce.c)
10.2 Program Data
Program Data (g01eyce.d)
10.3 Program Results
Program Results (g01eyce.r)