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

TLDR

Goodness‑of‑fit tests based on Cramér–von Mises and Anderson–Darling are considered for composite hypotheses where the distribution parameters are unknown, requiring a new critical region for each hypothesized distribution. The authors propose a transformation that yields a test statistic independent of the unknown parameters, eliminating the need to recompute critical regions. The transformation uses three asymptotically derived coefficients, tabulated once for several extreme‑value and related models with maximum‑likelihood‑estimated parameters, and its performance is evaluated via Monte‑Carlo simulations for small samples and power comparison.

Abstract

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, F H , whose parameters are unspecified. When this is the case, the critical region of the test has to be redetermined for each hypothetical distribution F H . To avoid this difficulty, a transformation is proposed that produces a new test statistic which is independent of F H . 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.