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

Abstract

Irrespective of the statistical model under study, the derivation of limits, in the Le Cam sense, of sequences of local experiments (see, e.g., Jeganathan, Econometric Theory 11:818–887, 1995 and Strasser, Mathematical Theory of Statistics: Statistical experiments and asymptotic decision theory, Walter de Gruyter, Berlin, 1985) often follows along very similar lines, essentially involving differentiability in quadratic mean of square roots of (conditional) densities. This chapter establishes two abstracts but quite generally applicable 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 estimators. Such results have been established, for locally asymptotically normal (LAN) models involving independent and identically distributed observations, by, e.g., Bickel et al. (Efficient and adaptive Estimation for semiparametric Models, Johns Hopkins University Press, Baltimore, 1993), van der Vaart (Statistical Estimation in Large Parameter Spaces, CWI, Amsterdam, 1988; Asymptotic Statistics, Cambridge University Press, Cambridge, 2000). Similar results are provided here for models exhibiting serial dependencies which, so far, have been treated on a case-by-case basis (see Hallin and Paindaveine, Journal of Statistical Planning and Inference 136:1–32, 2005 and Hallin and Puri, Journal of Multivariate Analysis 50:175–237, 1994 for typical examples) and, in general, under stronger regularity assumptions. Unlike their i.i.d. counterparts, our results are established under LAQ conditions, hence extend beyond the context of LAN experiments, so that nonstationary unit-root time series and cointegration models, for instance, also can be handled (see Hallin et al., Optimal pseudo-Gaussian and rank-based tests of the cointegrating rank in semiparametric error-correction models, 2013).