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THE GEOMETRIC STRUCTURE OF THE

EXPECTED/OBSERVED LIKELIHOOD EXPANSIONS

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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.

**Source**
( TeX ,
DVI ,
PS )