The geometric structure of the expected/observed likelihood expansions

Abstract

Stochastic expansions of likelihood quantities are a basic tool for asymptotic inference. The traditional derivation is through ordinary Taylor ex- pansions, rearranging terms according to their asymptotic order. The resulting expansions are called here ezpected/observed, being expressed in terms of the score vector, the expected information matrix, log likelihood derivatives and their joint moments. Though very convenient for many statistical purposes, ex- pected/observed expansions are not usually written in tensorial form. Recently, within a differential geometric approach to asymptotic statistical calculations, invariant Taylor expansions based on likelihood yokes have been introduced. The resulting formulae are invariant, but the quantities involved are in some respects less convenient for statistical purposes. The aim of this paper is to show that, through an invariant Taylor expansion of the coordinates related to the expected likelihood yoke, expected/observed expansions up to the fourth asymptotic order may be re-obtained from invariant Taylor expansions. This derivation produces invariant ezpected/observed expansions. Let jc = {p~ : co Eft c_ ~a} be a parametric family of probability distributions defined on a sample space/12 and dominated by a a-finite measure #. The param- eter space ~t is assumed to be an open non-empty subset of ~a. Let us denote by p(x; co), x ~ 2(, the density of P~ with respect to # and by l(a:) = logp(x; co) the log likelihood function based on the sample data x. We assume that the log likeli- hood function is a smooth function of the parameter and that the usual additional regularity conditions hold ensuring, in particular, that the maximum likelihood * This research was partially supported by the Italian National Research Council grant