# NAG Library Routine Document

## 1Purpose

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

## 2Specification

Fortran Interface
 Function g01eyf ( n, d,
 Real (Kind=nag_wp) :: g01eyf Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: d
#include <nagmk26.h>
 double g01eyf_ (const Integer *n, const double *d, Integer *ifail)

## 3Description

Let ${S}_{n}\left(x\right)$ be the sample cumulative distribution function and ${F}_{0}\left(x\right)$ the hypothesised theoretical distribution function.
g01eyf returns the upper tail probability, $p$, associated with the one-sided Kolmogorov–Smirnov test statistic ${D}_{n}^{+}$ or ${D}_{n}^{-}$, where these one-sided statistics are defined as follows;
 $Dn+ = supxSnx-F0x, Dn- = supxF0x-Snx.$
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).

## 4References

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

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of observations in the sample.
Constraint: ${\mathbf{n}}\ge 1$.
2:     $\mathbf{d}$ – Real (Kind=nag_wp)Input
On entry: contains the test statistic, ${D}_{n}^{+}$ or ${D}_{n}^{-}$.
Constraint: $0.0\le {\mathbf{d}}\le 1.0$.
3:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value  is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{d}}<0.0$ or ${\mathbf{d}}>1.0$: ${\mathbf{d}}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

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.

## 8Parallelism and Performance

g01eyf is not threaded in any implementation.

The upper tail probability for the two-sided 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 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 ${D}_{n}$ is good for small probabilities, (e.g., $p\le 0.10$) but becomes very poor for larger probabilities.
The time taken by the 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 $n$.

## 10Example

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.

### 10.1Program Text

Program Text (g01eyfe.f90)

### 10.2Program Data

Program Data (g01eyfe.d)

### 10.3Program Results

Program Results (g01eyfe.r)