Electronic Journal of Statistics

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

Claire Lacour

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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

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

First available in Project Euclid: 16 January 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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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