Electronic Journal of Statistics

Least squares type estimation of the transition density of a particular hidden Markov chain

Claire Lacour

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Abstract

In this paper, we study the following model of hidden Markov chain: Yi=Xi+ɛi, i=1,,n+1 with (Xi) a real-valued stationary Markov chain and (ɛi)1in+1 a noise having a known distribution and independent of the sequence (Xi). We present an estimator of the transition density obtained by minimization of an original contrast that takes advantage of the regressive aspect of the problem. It is selected among a collection of projection estimators with a model selection method. The L2-risk and its rate of convergence are evaluated for ordinary smooth noise and some simulations illustrate the method. We obtain uniform risk bounds over classes of Besov balls. In addition our estimation procedure requires no prior knowledge of the regularity of the true transition. Finally, our estimator permits to avoid the drawbacks of quotient estimators.

Article information

Source
Electron. J. Statist., Volume 2 (2008), 1-39.

Dates
First available in Project Euclid: 16 January 2008

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1200512011

Digital Object Identifier
doi:10.1214/07-EJS111

Mathematical Reviews number (MathSciNet)
MR2369084

Zentralblatt MATH identifier
1135.62064

Subjects
Primary: 62G05: Estimation
Secondary: 62M05: Markov processes: estimation 62H12: Estimation

Keywords
Hidden Markov Chain Transition Density Nonparametric Estimation Model Selection Rate of convergence

Citation

Lacour, Claire. Least squares type estimation of the transition density of a particular hidden Markov chain. Electron. J. Statist. 2 (2008), 1--39. doi:10.1214/07-EJS111. https://projecteuclid.org/euclid.ejs/1200512011


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