Performs a Kolmogorov-Smirnov two-sample test.

The type double function is imsls_d_kolmogorov_two.

Pointer to an array of length 3 containing Z, p1, and p2.

Function imsls_f_kolmogorov_two computes Kolmogorov-Smirnov two-sample test statistics for testing that two continuous cumulative distribution functions (CDF’s) are identical based upon two random samples. One- or two-sided alternatives are allowed. Exact p‑values are computed for the two-sided test when n_observations_x × n_observations_y is less than 104.

Let Fn(x) denote the empirical CDF in the X sample, let Gm(y) denote the empirical CDF in the Y sample, where n = n_observations_x - n_missing_x and m = n_observations_y - n_missing_y, and let the corresponding population distribution functions be denoted by F(x) and G(y), respectively. Then, the hypotheses tested by imsls_f_kolmogorov_two are as follows:

The test statistics are given as follows:

Asymptotically, the distribution of the statistic

(returned in test_statistics[0]) converges to a distribution given by Smirnov (1939).

Exact probabilities for the two-sided test are computed when n*m is less than or equal to 104, according to an algorithm given by Kim and Jennrich (1973;). When n*m is greater than 104, the very good approximations given by Kim and Jennrich are used to obtain the two-sided p-values. The one-sided probability is taken as one half the two-sided probability. This is a very good approximation when the p-value is small (say, less than 0.10) and not very good for large p‑values.

This example illustrates the imsls_f_kolmogorov_two routine with two randomly generated samples from a uniform(0,1) distribution. Since the two theoretical distributions are identical, we would not expect to reject the null hypothesis.