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: $d$ – doubleInput: On entry: contains the test statistic, $D_{n}^{+}$ or $D_{n}^{-}$ .

Constraint: $0.0 \leq d \leq 1.0$ .
3: $fail$ – NagError *Input/Output: The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .
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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL: On entry, $d < 0.0$ or $d > 1.0$ : $d = 〈value〉$ .

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.

nag_prob_1_sample_ks (g01eyc) is not threaded in any implementation.

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 nag_prob_1_sample_ks (g01eyc), that is

2 p

. (Note that if the probability from nag_prob_1_sample_ks (g01eyc) 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 function 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

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.