NAG Library Function Document

nag_prob_1_sample_ks (g01eyc)

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

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 3.6 in the Essential Introduction).

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.
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.

Not applicable.

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.