Documentation

Kolmogorov-Smirnov statistic for a set of data relative to a (continuous) distribution with the given CDF. Returns 3 common forms of the statistic: (K+, K-, D), where K+ and K- are Smirnov's one-sided forms as presented in Knuth's Semi-Numerical Algorithms (TAOCP, vol. 2) and D is Kolmogorov's undirected version.

In particular,

• K+ = sup(x -> F_n(x) - F(x)) * K- = sup(x -> F(x) - F_n(x)) * D = sup(x -> abs(F_n(x) - F(x)))

ksTest cdf xs Computes the probability of a random data set (of the same size as xs) drawn from a continuous distribution with the given CDF having the same Kolmogorov statistic as xs.

The statistic is the greatest absolute deviation of the empirical CDF of XS from the assumed CDF cdf.

If the data were, in fact, drawn from a distribution with the given CDF, then the resulting p-value should be uniformly distributed over (0,1].

KS distribution: not really a standard mathematical concept, but still a nice conceptual shift. KS n d is the distribution of a random variable constructed as a list of n independent random variables of distribution d.

The corresponding CDF instance implements the K-S test for such lists. For example, if xs is a list of length 100 believed to contain Beta(2,5) variates, then cdf (KS 100 (Beta 2 5)) is the K-S test for that distribution. (Note that if length xs is not 100, then the result will be 0 because such lists cannot arise from that KS distribution. Somewhat arbitrarily, all lists of "impossible" length are grouped at the bottom of the ordering encoded by the CDF instance.)

The KS test can easily be applied recursively. For example, if d is a Distribution of interest and you have 100 trials each with 100 data points, you can test it by calling cdf (KS 100 (KS 100 d)).