Data Driven Likelihood Ratio Tests for Goodness-of-Fit with Estimated Parameters

This paper generalizes the goodness of fit tests of Claeskens and Hjort (2004) and Marsh (2006) to the case where the hypothesis specifies only family of distributions. Data driven versions of these tests are based upon the Akaike and Bayesian selection criteria. The asymptotic distributions of these tests are shown to be standard, unlike those based upon the empirical distribution function. Moreover, numerical evidence suggests that under the null hypothesis performance is very similar to tests such as the Kolmogorov-Smirnov or Anderson-Darling. However, in terms of power under the alternative, the proposed tests seem to have a consistent and significant advantage.