NAG Library Function Document

nag_prob_2_sample_ks (g01ezc)

+− 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

nag_prob_2_sample_ks (g01ezc) returns the probability associated with the upper tail of the Kolmogorov–Smirnov two sample distribution.

2 Specification

#include <nag.h>

#include <nagg01.h>

double

nag_prob_2_sample_ks (Integer n1, Integer n2, double d, NagError *fail)

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 nag_prob_2_sample_ks (g01ezc) 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 Arguments

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 – doubleInput: 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: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_CONVERGENCE: The Smirnov approximation used for large samples did not converge in $200$ iterations. The probability is set to $1.0$ .
NE_INT: On entry, $n1 = ⟨value⟩$ and $n2 = ⟨value⟩$ .
Constraint: $n1 \geq 1$ and $n2 \geq 1$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL: On entry, $d < 0.0$ or $d > 1.0$ : $d = ⟨value⟩$ .

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 Parallelism and Performance

Not applicable.

9 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 nag_prob_2_sample_ks (g01ezc), 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 function 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}

10 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 Function Documentnag_prob_2_sample_ks (g01ezc)