nag_anderson_darling_stat (g08chc) calculates the Anderson–Darling goodness-of-fit test statistic.
the Anderson–Darling test statistic for
of a variable
assumed to be standard uniform and sorted in ascending order, then:
When observations of a random variable
are non-uniformly distributed, the probability integral transformation (PIT):
is the cumulative distribution function of the distribution of interest, yields a uniformly distributed random variable
. The PIT is true only if all parameters of a distribution are known as opposed to estimated; otherwise it is an approximation.
Anderson T W and Darling D A (1952) Asymptotic theory of certain ‘goodness-of-fit’ criteria based on stochastic processes Annals of Mathematical Statistics 23 193–212