Cramer-von Mises and Anderson-Darling goodness of fit tests for extreme value distributions with unknown parameters

The use of goodness of fit tests based on Cramer–von Mises and Anderson-Darling statistics is discussed, with reference to the composite hypothesis that a sample of observations comes from a distribution, FH, whose parameters are unspecified. When this is the case, the critical region of the test has to be redetermined for each hypothetical distribution FH. To avoid this difficulty, a transformation is proposed that produces a new test statistic which is independent of FH. This transformation involves three coefficients that are determined using the asymptotic theory of tests based on the empirical distribution function. A single table of coefficients is thus sufficient for carrying out the test with different hypothetical distributions; a set of probability models of common use in extreme value analysis is considered here, including the following: extreme value 1 and 2, normal and lognormal, generalized extreme value, three-parameter gamma, and log-Pearson type 3, in all cases with parameters estimated using maximum likelihood. Monte Carlo simulations are used to determine small sample corrections and to assess the power of the tests compared to alternative approaches.