On Quadratic Expansions of Log-Likelihoods and a General Asymptotic Linearity Result

Irrespective of the statistical model under study, the derivation of limits,in the Le Cam sense, of sequences of local experiments (see [7]-[10]) oftenfollows along very similar lines, essentially involving differentiability in quadraticmean of square roots of (conditional) densities. This chapter establishes two abstractand very general results providing sufficient and nearly necessary conditionsfor (i) the existence of a quadratic expansion, and (ii) the asymptotic linearity oflocal log-likelihood ratios (asymptotic linearity is needed, for instance, when unspecifiedmodel parameters are to be replaced, in some statistic of interest, withsome preliminary estimator). Such results have been established, for locally asymptoticallynormal (LAN) models involving independent and identically distributedobservations, by, e.g. [1], [11] and [12]. Similar results are provided here for modelsexhibiting serial dependencies which, so far, have been treated on a case-by-casebasis (see [4] and [5] for typical examples) and, in general, under stronger regularityassumptions. Unlike their i.i.d. counterparts, our results extend beyond the contextof LAN experiments, so that non-stationary unit-root time series and cointegrationmodels, for instance, also can be handled (see [6]).