NAG Library Routine Document
G01EZF returns the probability associated with the upper tail of the Kolmogorov–Smirnov two sample distribution, via the routine name.
|REAL (KIND=nag_wp) G01EZF
||N1, N2, IFAIL
Let and denote the empirical cumulative distribution functions for the two samples, where and are the sizes of the first and second samples respectively.
The function G01EZF computes the upper tail probability for the Kolmogorov–Smirnov two sample two-sided test statistic
The probability is computed exactly if
using a method given by Kim and Jenrich (1973)
. For the case where
the Smirnov approximation is used. For all other cases the Kolmogorov approximation is used. These two approximations are discussed in Kim and Jenrich (1973)
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 Selected Tables in Mathematical Statistics 1 80–129 American Mathematical Society
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: N1 – INTEGERInput
On entry: the number of observations in the first sample, .
- 2: N2 – INTEGERInput
On entry: the number of observations in the second sample, .
- 3: D – REAL (KIND=nag_wp)Input
On entry: the test statistic , for the two sample Kolmogorov–Smirnov goodness-of-fit test, that is the maximum difference between the empirical cumulative distribution functions (CDFs) of the two samples.
- 4: IFAIL – INTEGERInput/Output
must be set to
. If you are unfamiliar with this parameter you should refer to Section 3.3
in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
The approximation solution did not converge in iterations. A tail probability of is returned by G01EZF.
The large sample distributions used as approximations to the exact distribution should have a relative error of less than 5% for most cases.
The upper tail probability for the one-sided statistics, or , can be approximated by halving the two-sided upper tail probability returned by G01EZF, that is . This approximation to the upper tail probability for either or is good for small probabilities, (e.g., ) but becomes poor for larger probabilities.
The time taken by the routine increases with and , until or . At this point one of the approximations is used and the time decreases significantly. The time then increases again modestly with and .
The following example reads in different sample sizes and values for the test statistic . The upper tail probability is computed and printed for each case.
9.1 Program Text
Program Text (g01ezfe.f90)
9.2 Program Data
Program Data (g01ezfe.d)
9.3 Program Results
Program Results (g01ezfe.r)