Abstract.
This paper is devoted to the problem of estimating functionals of type μ(f)=∫fdμ from observations drawn from a positive recurrent atomic Markov chain with stationary distribution μ. The properties of different estimators are studied. Beyond an accurate estimation of their bias, the estimation of their asymptotic variance is considered. We also show that the results of Malinovskii(1987) on the validity of the formal Edgeworth expansion for sample mean statistics of type T n = n−1∑n i=1 f(X i ) extend to their studentized versions, normalized by the asymptotic variance estimates we consider.
Resume.
Cet article est consacré au problème de l’estimation d’une fonctionnelle linéaire μ(f)=∫fdμ à partir de l’observation d’une chaîne de Markov récurrente positive possédant un atome accessible et de distribution stationnaire μ. Les propriétés de plusieurs estimateurs sont étudiées. Au delà d’une estimation précise de leurs biais respectifs, nous nous intéressons également à l’estimation de la variance asymptotique de ces estimateurs. Nous montrons aussi que les résultats de Malinovskii (1987) concernant le développement d’Edgeworth de l’estimateur T n = n−1∑n i =1f(X i ) s’étendent à la version studentisée, lorsqu’il est normalisé par l’estimateur de la variance asymptotique que nous proposons.
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Revised version: 3 March 2004
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Bertail, P., Clémençon, S. Edgeworth expansions of suitably normalized sample mean statistics for atomic Markov chains. Probab. Theory Relat. Fields 130, 388–414 (2004). https://doi.org/10.1007/s00440-004-0360-0
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DOI: https://doi.org/10.1007/s00440-004-0360-0