NAG Library Function Document
nag_prob_1_sample_ks (g01eyc) returns the upper tail probability associated with the one sample Kolmogorov–Smirnov distribution.
||nag_prob_1_sample_ks (Integer n,
Let be the sample cumulative distribution function and the hypothesised theoretical distribution function.
nag_prob_1_sample_ks (g01eyc) returns the upper tail probability,
, associated with the one-sided Kolmogorov–Smirnov test statistic
, where these one-sided statistics are defined as follows;
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)
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
n – IntegerInput
On entry: , the number of observations in the sample.
d – doubleInput
On entry: contains the test statistic, or .
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
On entry, .
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
On entry, or : .
The large sample distribution used as an approximation to the exact distribution should have a relative error of less than % for most cases.
8 Parallelism and Performance
The upper tail probability for the two-sided statistic, , can be approximated by twice the probability returned via nag_prob_1_sample_ks (g01eyc), that is . (Note that if the probability from nag_prob_1_sample_ks (g01eyc) is greater than then the two-sided probability should be truncated to ). This approximation to the tail probability for is good for small probabilities, (e.g., ) but becomes very poor for larger probabilities.
The time taken by the function increases with , until . At this point the approximation is used and the time decreases significantly. The time then increases again modestly with .
The following example reads in different sample sizes and values for the test statistic . The upper tail probability is computed and printed for each case.
10.1 Program Text
Program Text (g01eyce.c)
10.2 Program Data
Program Data (g01eyce.d)
10.3 Program Results
Program Results (g01eyce.r)