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Adaptive estimation of the transition density of a particular hidden Markov chain

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  • Lacour, Claire

Abstract

We study the following model of hidden Markov chain: with (Xi) a real-valued positive recurrent and stationary Markov chain, and ([var epsilon]i)1[less-than-or-equals, slant]i[less-than-or-equals, slant]n+1 a noise independent of the sequence (Xi) having a known distribution. We present an adaptive estimator of the transition density based on the quotient of a deconvolution estimator of the density of Xi and an estimator of the density of (Xi,Xi+1). These estimators are obtained by contrast minimization and model selection. We evaluate the L2 risk and its rate of convergence for ordinary smooth and supersmooth noise with regard to ordinary smooth and supersmooth chains. Some examples are also detailed.

Suggested Citation

  • Lacour, Claire, 2008. "Adaptive estimation of the transition density of a particular hidden Markov chain," Journal of Multivariate Analysis, Elsevier, vol. 99(5), pages 787-814, May.
    • Handle: RePEc:eee:jmvana:v:99:y:2008:i:5:p:787-814
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    1. Chaleyat-Maurel, Mireille & Genon-Catalot, Valentine, 2006. "Computable infinite-dimensional filters with applications to discretized diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 116(10), pages 1447-1467, October.
    2. Leroux, Brian G., 1992. "Maximum-likelihood estimation for hidden Markov models," Stochastic Processes and their Applications, Elsevier, vol. 40(1), pages 127-143, February.
    3. Stefanski, Leonard A., 1990. "Rates of convergence of some estimators in a class of deconvolution problems," Statistics & Probability Letters, Elsevier, vol. 9(3), pages 229-235, March.
    4. Masry, Elias, 1993. "Strong consistency and rates for deconvolution of multivariate densities of stationary processes," Stochastic Processes and their Applications, Elsevier, vol. 47(1), pages 53-74, August.
    5. C.C.Y. Dorea & L.C. Zhao, 2002. "Nonparametric Density Estimation in Hidden Markov Models," Statistical Inference for Stochastic Processes, Springer, vol. 5(1), pages 55-64, January.
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    1. Gaëlle Chagny & Claire Lacour, 2015. "Optimal adaptive estimation of the relative density," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(3), pages 605-631, September.
    2. Gautier, Eric & Gaillac, Christophe, 2019. "Adaptive estimation in the linear random coefficients model when regressors have limited variation," TSE Working Papers 19-1026, Toulouse School of Economics (TSE).
    3. Salima El Kolei & Fabien Navarro, 2022. "Contrast estimation for noisy observations of diffusion processes via closed-form density expansions," Statistical Inference for Stochastic Processes, Springer, vol. 25(2), pages 303-336, July.
    4. Sandra Plancade, 2011. "Model selection for hazard rate estimation in presence of censoring," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 74(3), pages 313-347, November.
    5. Gaëlle Chagny, 2015. "Adaptive Warped Kernel Estimators," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(2), pages 336-360, June.

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