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Stochastics
An International Journal of Probability and Stochastic Processes
Volume 80, 2008 - Issue 2-3: A Festschrift for Priscilla Greenwood
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Original Articles

Optimality of estimators for misspecified semi-Markov models

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Pages 181-196 | Received 24 Jul 2007, Accepted 07 Nov 2007, Published online: 10 Oct 2008
 

Abstract

Suppose we observe a geometrically ergodic semi-Markov process and have a parametric model for the transition distribution of the embedded Markov chain, for the conditional distribution of the inter-arrival times, or for both. The first two models for the process are semiparametric, and the parameters can be estimated by conditional maximum likelihood estimators. The third model for the process is parametric, and the parameter can be estimated by an unconditional maximum likelihood estimator. We determine heuristically the asymptotic distributions of these estimators and show that they are asymptotically efficient. If the parametric models are not correct, the (conditional) maximum likelihood estimators estimate the parameter that maximizes the Kullback–Leibler information. We show that they remain asymptotically efficient in a nonparametric sense.

AMS Subject Classification:

Acknowledgement

Anton Schick was suupported in part by NSF Grant DMS0405791

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