THE GEOMETRIC STRUCTURE OF THE EXPECTED/OBSERVED LIKELIHOOD EXPANSIONS

LUIGI PACE AND ALESSANDRA SALVAN

Department of Statistics, University of Padova, Via S.Francesco 33, I-35121 Padova, Italy

(Received June 21, 1993; revised March 22, 1994)

Abstract. Stochastic expansions of likelihood quantities are a basic tool for asymptotic inference. The traditional derivation is through ordinary Taylor expansions, rearranging terms according to their asymptotic order. The resulting expansions are called here expected/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, expected/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 expected/observed expansions.

Key words and phrases: Asymptotic expansions, index notation, invariant Taylor series expansions, likelihood, tensors, yokes.