G01 Chapter Contents
G01 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG01EYF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

G01EYF returns the upper tail probability associated with the one sample Kolmogorov–Smirnov distribution, via the routine name.

## 2  Specification

 FUNCTION G01EYF ( N, D, IFAIL)
 REAL (KIND=nag_wp) G01EYF
 INTEGER N, IFAIL REAL (KIND=nag_wp) D

## 3  Description

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).

## 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  Parameters

1:     N – INTEGERInput
On entry: $n$, the number of observations in the sample.
Constraint: ${\mathbf{N}}\ge 1$.
2:     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:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. 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 $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ 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).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, 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, ${\mathbf{N}}<1$.
${\mathbf{IFAIL}}=2$
 On entry, ${\mathbf{D}}<0.0$, or ${\mathbf{D}}>1.0$.

## 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.

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 $n$.

## 9  Example

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.

### 9.1  Program Text

Program Text (g01eyfe.f90)

### 9.2  Program Data

Program Data (g01eyfe.d)

### 9.3  Program Results

Program Results (g01eyfe.r)