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NAG Library Routine Document

g01eyf (prob_kolmogorov1)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

g01eyf returns the upper tail probability associated with the one sample Kolmogorov–Smirnov distribution, via the routine name.

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 nagmk26.h

double

g01eyf_ ( const Integer *n, const double *d, Integer *ifail)

3

Description

Let

S_{n} (x)

be the sample cumulative distribution function and

F_{0} (x)

the hypothesised theoretical distribution function.

g01eyf returns the upper tail probability,

p

, associated with the one-sided Kolmogorov–Smirnov test statistic

D_{n}^{+}

D_{n}^{-}

, where these one-sided statistics are defined as follows;

\begin{array}{lcl} D_{n}^{+} & = & \sup_{x} [S_{n} (x) - F_{0} (x)], \\ D_{n}^{-} & = & \sup_{x} [F_{0} (x) - S_{n} (x)[. \end{array}

n \leq 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

Arguments

1: $n$ – IntegerInput: On entry: $n$ , the number of observations in the sample.

Constraint: $n \geq 1$ .
2: $d$ – Real (Kind=nag_wp)Input: On entry: contains the test statistic, $D_{n}^{+}$ or $D_{n}^{-}$ .

Constraint: $0.0 \leq d \leq 1.0$ .
3: $ifail$ – IntegerInput/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . 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 $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, 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, $n = 〈value〉$ .
Constraint: $n \geq 1$ .

$ifail = 2$: On entry, $d < 0.0$ or $d > 1.0$ : $d = 〈value〉$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation 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

g01eyf is not threaded in any implementation.

9

Further Comments

The upper tail probability for the two-sided statistic,

D_{n} = \max (D_{n}^{+}, D_{n}^{-})

, can be approximated by twice the probability returned via g01eyf, that is

2 p

. (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 \leq 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

10

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.

NAG Library Routine Document

g01eyf (prob_kolmogorov1)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results