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

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  • Marc Hallin
  • Ramon van den Akker
  • Bas Werker

Abstract

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]) 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]).

Suggested Citation

  • Marc Hallin & Ramon van den Akker & Bas Werker, 2013. "On Quadratic Expansions of Log-Likelihoods and a General Asymptotic Linearity Result," Working Papers ECARES ECARES 2013-34, ULB -- Universite Libre de Bruxelles.
  • Handle: RePEc:eca:wpaper:2013/149099
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    References listed on IDEAS

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    1. Marc Hallin & Abdelhadi Akharif, 2003. "Efficient detection of random coefficients in AR(p) models," ULB Institutional Repository 2013/2121, ULB -- Universite Libre de Bruxelles.
    2. Abdelhadi Akharif & Marc Hallin, 2003. "Efficient detection of random coefficients in autoregressive models," ULB Institutional Repository 2013/127956, ULB -- Universite Libre de Bruxelles.
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    Cited by:

    1. Xiaohong Chen & Timothy Christensen & Keith O'Hara & Elie Tamer, 2016. "MCMC Confidence sets for Identified Sets," Cowles Foundation Discussion Papers 2037R, Cowles Foundation for Research in Economics, Yale University, revised Jul 2016.
    2. Paindaveine, Davy & Van Bever, Germain, 2014. "Inference on the shape of elliptical distributions based on the MCD," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 125-144.
    3. Hallin, M. & Werker, B.J.M. & van den Akker, R., 2015. "Optimal Pseudo-Gaussian and Rank-based Tests of the Cointegration Rank in Semiparametric Error-correction Models," Discussion Paper 2015-001, Tilburg University, Center for Economic Research.

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