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:
n – IntegerInput
On entry: $n$, the number of observations in the sample.
Constraint:
${\mathbf{n}}\ge 1$.
 2:
d – doubleInput
On entry: contains the test statistic, ${D}_{n}^{+}$ or ${D}_{n}^{}$.
Constraint:
$0.0\le {\mathbf{d}}\le 1.0$.
 3:
fail – NagError *Input/Output

The NAG error argument (see
Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
 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.
 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.
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 $n$.
9 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.
9.1 Program Text
Program Text (g01eyce.c)
9.2 Program Data
Program Data (g01eyce.d)
9.3 Program Results
Program Results (g01eyce.r)