On quadratic expansions of log-likelihoods and a general asymptotic linearity result

Irrespective of the statistical model under study, the derivation of lim- its, 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 ab- stract 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 un- specified model parameters are to be replaced, in some statistic of interest, with some preliminary estimator). Such results have been established, for locally asymp- totically normal (LAN) models involving independent and identically distributed observations, by, e.g., [1], [11] and [12]. Similar results are provided here for mod- els 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]).