NAG FL Interface
g01ezf (prob_​kolmogorov2)

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1 Purpose

g01ezf returns the probability associated with the upper tail of the Kolmogorov–Smirnov two sample distribution.

2 Specification

Fortran Interface
Function g01ezf ( n1, n2, d, ifail)
Real (Kind=nag_wp) :: g01ezf
Integer, Intent (In) :: n1, n2
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: d
C Header Interface
#include <nag.h>
double  g01ezf_ (const Integer *n1, const Integer *n2, const double *d, Integer *ifail)
The routine may be called by the names g01ezf or nagf_stat_prob_kolmogorov2.

3 Description

Let Fn1(x) and Gn2(x) denote the empirical cumulative distribution functions for the two samples, where n1 and n2 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 Dn1,n2, where
The probability is computed exactly if n1,n210000 and max(n1,n2)2500 using a method given by Kim and Jenrich (1973). For the case where min(n1,n2) 10 % of the max(n1,n2) and min(n1,n2) 80 the Smirnov approximation is used. For all other cases the Kolmogorov approximation is used. These two approximations are discussed in Kim and Jenrich (1973).

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
Kim P J and Jenrich R I (1973) Tables of exact sampling distribution of the two sample Kolmogorov–Smirnov criterion Dmn(m<n) 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

5 Arguments

1: n1 Integer Input
On entry: the number of observations in the first sample, n1.
Constraint: n11.
2: n2 Integer Input
On entry: the number of observations in the second sample, n2.
Constraint: n21.
3: d Real (Kind=nag_wp) Input
On entry: the test statistic Dn1,n2, 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.
Constraint: 0.0d1.0.
4: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry 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:
On entry, n1=value and n2=value.
Constraint: n11 and n21.
On entry, d<0.0 or d>1.0: d=value.
The Smirnov approximation used for large samples did not converge in 200 iterations. The probability is set to 1.0.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The large sample distributions used as approximations to the exact distribution should have a relative error of less than 5% for most cases.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g01ezf is not threaded in any implementation.

9 Further Comments

The upper tail probability for the one-sided statistics, Dn1,n2+ or Dn1,n2-, can be approximated by halving the two-sided upper tail probability returned by g01ezf, that is p/2. This approximation to the upper tail probability for either Dn1,n2+ or Dn1,n2- is good for small probabilities, (e.g., p0.10) but becomes poor for larger probabilities.
The time taken by the routine increases with n1 and n2, until n1n2>10000 or max(n1,n2)2500. At this point one of the approximations is used and the time decreases significantly. The time then increases again modestly with n1 and n2.

10 Example

The following example reads in 10 different sample sizes and values for the test statistic Dn1,n2. The upper tail probability is computed and printed for each case.

10.1 Program Text

Program Text (g01ezfe.f90)

10.2 Program Data

Program Data (g01ezfe.d)

10.3 Program Results

Program Results (g01ezfe.r)