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August, 1995 Explicit Limit Results for Minimal Sufficient Statistics and Maximum Likelihood Estimators in Some Markov Processes: Exponential Families Approach
Valeri T. Stefanov
Ann. Statist. 23(4): 1073-1101 (August, 1995). DOI: 10.1214/aos/1176324699

Abstract

Finite-state Markov chains with either a discrete or continuous time parameter, Markov renewal processes and Markov-additive processes are considered. We prove that their likelihood functions, in the nonsequential as well as in various sequential cases, belong to special $(n + k, n)$-curved exponential families in general, for which limit results are easily established. Subsequently, asymptotic normality of the corresponding nonsequential and sequential maximum likelihood estimators is established. Also in the case of Markov renewal and Markov-additive processes, stopping times are determined which reduce the corresponding curved exponential families in general to noncurved ones. The latter, together with results of Stefanov, are combined with results of Serfozo to imply explicit solutions in functional limit theorems for the considered processes. In particular, we derive explicit solutions for the important variance parameter in the functional central limit theorems and functional laws of iterated logarithm for those processes. Indeed, our explicit solutions cover more general cases than the known ones, even in the case of finite-state Markov chains. Moreover, we supply explicit solutions, not previously available, in functional limit theorems for Markov renewal processes and Markov-additive processes.

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Valeri T. Stefanov. "Explicit Limit Results for Minimal Sufficient Statistics and Maximum Likelihood Estimators in Some Markov Processes: Exponential Families Approach." Ann. Statist. 23 (4) 1073 - 1101, August, 1995. https://doi.org/10.1214/aos/1176324699

Information

Published: August, 1995
First available in Project Euclid: 11 April 2007

zbMATH: 0852.62076
MathSciNet: MR1353496
Digital Object Identifier: 10.1214/aos/1176324699

Subjects:
Primary: 62L99
Secondary: 60F17, 62M05

Rights: Copyright © 1995 Institute of Mathematical Statistics

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Vol.23 • No. 4 • August, 1995
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