NAG FL Interface
g01eyf (prob_​kolmogorov1)

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1 Purpose

g01eyf returns the upper tail probability associated with the one sample Kolmogorov–Smirnov distribution.

2 Specification

Fortran Interface
Function g01eyf ( n, d, ifail)
Real (Kind=nag_wp) :: g01eyf
Integer, Intent (In) :: n
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (In) :: d
C Header Interface
#include <nag.h>
double  g01eyf_ (const Integer *n, const double *d, Integer *ifail)
The routine may be called by the names g01eyf or nagf_stat_prob_kolmogorov1.

3 Description

Let Sn(x) be the sample cumulative distribution function and F0(x) the hypothesised theoretical distribution function.
g01eyf 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+ = supx[Sn(x)-F0(x)], Dn- = supx[F0(x)-Sn(x)[.  
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).

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) 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: n Integer Input
On entry: n, the number of observations in the sample.
Constraint: n1.
2: d Real (Kind=nag_wp) Input
On entry: contains the test statistic, Dn+ or Dn-.
Constraint: 0.0d1.0.
3: 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, n=value.
Constraint: n1.
On entry, d<0.0 or d>1.0: d=value.
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 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

Background information to multithreading can be found in the Multithreading documentation.
g01eyf is not threaded in any implementation.

9 Further Comments

The upper tail probability for the two-sided statistic, Dn=max(Dn+,Dn-), can be approximated by twice the probability returned via g01eyf, that is 2p. (Note that if the probability from g01eyf 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 routine 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 Dn. The upper tail probability is computed and printed for each case.

10.1 Program Text

Program Text (g01eyfe.f90)

10.2 Program Data

Program Data (g01eyfe.d)

10.3 Program Results

Program Results (g01eyfe.r)