NONNULL ASYMPTOTIC DISTRIBUTIONS OF 3 CLASSIC CRITERIA IN GENERALIZED LINEAR-MODELS

This paper develops simple formulae for the asymptotic expansions up to order n(-1/2) of the distributions of the likelihood ratio, Wald and score statistics in generalised linear models, under a sequence of Pitman alternatives. The powers of all three criteria, which are equivalent to first order, are compared under specific conditions based on second order approximations. The asymptotic distributions of all three statistics are obtained for testing a subset of linear parameters when the dispersion parameter is known, for testing the dispersion parameter and for testing the whole set of linear parameters assuming that the dispersion parameter is unknown. The formulae derived are simple enough to be used analytically to obtain closed-form expressions for these expansions in special models with a Fisher information matrix in closed form.