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

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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]).
Original languageEnglish
Title of host publicationMathematical Statistics and Limit Theorems
Subtitle of host publicationFestschrift in Honour of Paul Deheuvels
EditorsM. Hallin, D.M. Mason, D. Pfeifer, J. Steinebach
Place of PublicationCham
Number of pages19
ISBN (Electronic)9783319124421
ISBN (Print)9783319124414
Publication statusPublished - 8 Apr 2015

Publication series

NameSpringer Proceedings in Mathematics & Statistics

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    Hallin, M., van den Akker, R., & Werker, B. J. M. (2015). On quadratic expansions of log-likelihoods and a general asymptotic linearity result. In M. Hallin, D. M. Mason, D. Pfeifer, & J. Steinebach (Eds.), Mathematical Statistics and Limit Theorems : Festschrift in Honour of Paul Deheuvels (pp. 147-165). (Springer Proceedings in Mathematics & Statistics). Springer. https://doi.org/10.1007/978-3-319-12442-1_9