NAG Library Function Document

nag_prob_1_sample_ks (g01eyc)


    1  Purpose
    7  Accuracy


nag_prob_1_sample_ks (g01eyc) returns the upper tail probability associated with the one sample Kolmogorov–Smirnov distribution.


#include <nag.h>
#include <nagg01.h>
double  nag_prob_1_sample_ks (Integer n, double d, NagError *fail)


Let Snx be the sample cumulative distribution function and F0x the hypothesised theoretical distribution function.
nag_prob_1_sample_ks (g01eyc) returns the upper tail probability, p, associated with the one-sided Kolmogorov–Smirnov test statistic Dn+ or Dn-, where these one-sided statistics are defined as follows;
Dn+ = supxSnx-F0x, Dn- = supxF0x-Snx.  
If n100 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).


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) Non-parametric 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


1:     n IntegerInput
On entry: n, the number of observations in the sample.
Constraint: n1.
2:     d doubleInput
On entry: contains the test statistic, Dn+ or Dn-.
Constraint: 0.0d1.0.
3:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
On entry, n=value.
Constraint: n1.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
On entry, d<0.0 or d>1.0: d=value.


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.

Parallelism and Performance

nag_prob_1_sample_ks (g01eyc) is not threaded in any implementation.

Further Comments

The upper tail probability for the two-sided statistic, Dn=maxDn+,Dn-, 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 two-sided probability should be truncated to 1.0). This approximation to the tail probability for Dn is good for small probabilities, (e.g., p0.10) 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.


The following example reads in 10 different sample sizes and values for the test statistic Dn. The upper tail probability is computed and printed for each case.

Program Text

Program Text (g01eyce.c)

Program Data

Program Data (g01eyce.d)

Program Results

Program Results (g01eyce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017