(i) $H_{1}$ : $F (x) \neq G (x)$ .
(ii) $H_{2}$ : $F (x) > G (x)$ . This alternative hypothesis is sometimes stated as, ‘The $x$ 's tend to be smaller than the $y$ 's’, i.e., it would be demonstrated in practical terms if the values of $S_{1} (x)$ tended to exceed the corresponding values of $S_{2} (x)$ .
(iii) $H_{3}$ : $F (x) < G (x)$ . This alternative hypothesis is sometimes stated as, ‘The $x$ 's tend to be larger than the $y$ 's’, i.e., it would be demonstrated in practical terms if the values of $S_{2} (x)$ tended to exceed the corresponding values of $S_{1} (x)$ .

One of the following test statistics is computed depending on the particular alternative null hypothesis specified (see the description of the argument dtype in Section 5).

For the alternative hypothesis

H_{1}

$D_{n_{1}, n_{2}}$ – the largest absolute deviation between the two sample cumulative distribution functions.

For the alternative hypothesis

H_{2}

$D_{n_{1}, n_{2}}^{+}$ – the largest positive deviation between the sample cumulative distribution function of the first sample, $S_{1} (x)$ , and the sample cumulative distribution function of the second sample, $S_{2} (x)$ . Formally $D_{n_{1}, n_{2}}^{+} = \max \{S_{1} (x) - S_{2} (x), 0\}$ .

For the alternative hypothesis

H_{3}

$D_{n_{1}, n_{2}}^{-}$ – the largest positive deviation between the sample cumulative distribution function of the second sample, $S_{2} (x)$ , and the sample cumulative distribution function of the first sample, $S_{1} (x)$ . Formally $D_{n_{1}, n_{2}}^{-} = \max \{S_{2} (x) - S_{1} (x), 0\}$ .

g08cdc also returns the standardized statistic

Z = \sqrt{(n_{1} + n_{2} / n_{1} n_{2})} \times D

where

D

may be

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

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

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

depending on the choice of the alternative hypothesis. The distribution of this statistic converges asymptotically to a distribution given by Smirnov as

n_{1}

and

n_{2}

increase (see Feller (1948), Kendall and Stuart (1973), Kim and Jenrich (1973), Smirnov (1933) or Smirnov (1948)).

The probability, under the null hypothesis, of obtaining a value of the test statistic as extreme as that observed, is computed. If

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

and

n_{1} n_{2} \leq 10000

then an exact method given by Kim and Jenrich is used. Otherwise

p

is computed using the approximations suggested by Kim and Jenrich (see Kim and Jenrich (1973)). Note that the method used is only exact for continuous theoretical distributions. This method computes the two-sided probability. The one-sided probabilities are estimated by halving the two-sided probability. This is a good estimate for small

p

, that is

p \leq 0.10

, but it becomes very poor for larger

p

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 (1933) Estimate of deviation between empirical distribution functions in two independent samples Bull. Moscow Univ. 2(2) 3–16

Smirnov N (1948) Table for estimating the goodness of fit of empirical distributions Ann. Math. Statist. 19 279–281

5 Arguments

1: $n1$ – Integer Input

On entry: the number of observations in the first sample,

n_{1}

Constraint:

n1 \geq 1

2: $x [n1]$ – const double Input

On entry: the observations from the first sample,

x_{1}, x_{2}, \dots, x_{n_{1}}

3: $n2$ – Integer Input

On entry: the number of observations in the second sample,

n_{2}

Constraint:

n2 \geq 1

4: $y [n2]$ – const double Input

On entry: the observations from the second sample,

y_{1}, y_{2}, \dots, y_{n_{2}}

5: $dtype$ – Nag_TestStatistics Input

On entry: the statistic to be computed, i.e., the choice of alternative hypothesis.

$dtype = Nag_TestStatisticsDAbs$: Computes $D_{n_{1} n_{2}}$ , to test against $H_{1}$ .
$dtype = Nag_TestStatisticsDPos$: Computes $D_{n_{1} n_{2}}^{+}$ , to test against $H_{2}$ .
$dtype = Nag_TestStatisticsDNeg$: Computes $D_{n_{1} n_{2}}^{-}$ , to test against $H_{3}$ .

Constraint:

dtype = Nag_TestStatisticsDAbs

Nag_TestStatisticsDPos

Nag_TestStatisticsDNeg

6: $d$ – double * Output

On exit: the Kolmogorov–Smirnov test statistic (

D_{n_{1} n_{2}}

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

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

according to the value of dtype).

7: $z$ – double * Output

On exit: a standardized value,

Z

, of the test statistic,

D

, without any correction for continuity.

8: $p$ – double * Output

On exit: the tail probability associated with the observed value of

D

, where

D

may be

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

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

depending on the value of dtype (see Section 3).

9: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument dtype had an illegal value.
NE_G08CD_CONV: The iterative procedure used in the approximation of the probability for large n1 and n2 did not converge. For the two-sided test, $p = 1.0$ is returned. For the one-sided test, $p = 0.5$ is returned.
NE_INT_ARG_LT: On entry, n1 must not be less than 1: $n1 = 〈value〉$ .

On entry, n2 must not be less than 1: $n2 = 〈value〉$ .
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.

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

g08cdc is not threaded in any implementation.

9 Further Comments

The time taken by g08cdc 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 computes the two-sided Kolmogorov–Smirnov test statistic for two independent samples of size 100 and 50 respectively. The first sample is from a uniform distribution

U (0, 2)

. The second sample is from a uniform distribution

U (0.25, 2.25)

. The test statistic,

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

, the standardized test statistic,

Z

, and the tail probability,

p

, are computed and printed.

g08cd: FL CL CPP AD

NAG CL Interfaceg08cdc (test_​ks_​2sample)

▸▿ 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

NAG CL Interface
g08cdc (test_ks_2sample)