NAG Library Routine Document

G01EZF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G01EZF returns the probability associated with the upper tail of the Kolmogorov–Smirnov two sample distribution, via the routine name.

2 Specification

FUNCTION G01EZF (

N1, N2, D, IFAIL)

REAL (KIND=nag_wp) G01EZF

INTEGER	N1, N2, IFAIL
REAL (KIND=nag_wp)	D

3 Description

Let

F_{n_{1}} (x)

and

G_{n_{2}} (x)

denote the empirical cumulative distribution functions for the two samples, where

n_{1}

and

n_{2}

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

D_{n_{1}, n_{2}}

, where

D_{n_{1}, n_{2}} = \sup_{x} |F_{n_{1}} (x) - G_{n_{2}} (x)| .

The probability is computed exactly if

n_{1}, n_{2} \leq 10000

and

\max (n_{1}, n_{2}) \leq 2500

using a method given by Kim and Jenrich (1973). For the case where

\min (n_{1}, n_{2}) \leq 10 %

of the

\max (n_{1}, n_{2})

and

\min (n_{1}, n_{2}) \leq 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

D_{m n} (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 Parameters

1: N1 – INTEGERInput: On entry: the number of observations in the first sample, $n_{1}$ .
Constraint: $N1 \geq 1$ .
2: N2 – INTEGERInput: On entry: the number of observations in the second sample, $n_{2}$ .
Constraint: $N2 \geq 1$ .
3: D – REAL (KIND=nag_wp)Input: On entry: the test statistic $D_{n_{1}, n_{2}}$ , 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: $0.0 \leq D \leq 1.0$ .
4: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . 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 $- 1 or 1$ 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 $- 1 or 1$ 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

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,	$N1 < 1$ ,
or	$N2 < 1$ .

$IFAIL = 2$

On entry,	$D < 0.0$ ,
or	$D > 1.0$ .

$IFAIL = 3$: The approximation solution did not converge in $500$ iterations. A tail probability of $1.0$ is returned by G01EZF.

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 Further Comments

The upper tail probability for the one-sided statistics,

D_{n_{1}, n_{2}}^{+}

D_{n_{1}, n_{2}}^{-}

, 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

D_{n_{1}, n_{2}}^{+}

D_{n_{1}, n_{2}}^{-}

is good for small probabilities, (e.g.,

p \leq 0.10

) but becomes poor for larger probabilities.

The time taken by the routine increases with

n_{1}

and

n_{2}

, until

n_{1} n_{2} > 10000

\max (n_{1}, n_{2}) \geq 2500

. At this point one of the approximations is used and the time decreases significantly. The time then increases again modestly with

n_{1}

and

n_{2}

9 Example

The following example reads in

10

different sample sizes and values for the test statistic

D_{n_{1}, n_{2}}

. The upper tail probability is computed and printed for each case.

NAG Library Routine DocumentG01EZF