On quadratic expansions of log likelihoods and a general asymptotic linearity result
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Date
2013-09-17
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Abstract
Irrespective of the statistical model under study, the derivation of limits,
in the Le Cam sense, of sequences of local experiments (see [7]-[10]) often
follows along very similar lines, essentially involving differentiability in quadratic
mean of square roots of (conditional) densities. This chapter establishes two abstract
and very general results providing sufficient and nearly necessary conditions
for (i) the existence of a quadratic expansion, and (ii) the asymptotic linearity of
local log-likelihood ratios (asymptotic linearity is needed, for instance, when unspecified
model parameters are to be replaced, in some statistic of interest, with
some preliminary estimator). Such results have been established, for locally asymptotically
normal (LAN) models involving independent and identically distributed
observations, by, e.g., [1], [11] and [12]. Similar results are provided here for models
exhibiting serial dependencies which, so far, have been treated on a case-by-case
basis (see [4] and [5] for typical examples) and, in general, under stronger regularity
assumptions. Unlike their i.i.d. counterparts, our results extend beyond the context
of LAN experiments, so that non-stationary unit-root time series and cointegration
models, for instance, also can be handled (see [6]).