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

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

2006-11-22

arxiv.org/abs/math/0611681v1

Journal of Multivariate Analysis 99, 5 (2008) 787-814

Mathematics

Statistics Theory

Scientific paper

10.1016/j.jmva.2007.04.006

We study the following model of hidden Markov chain: $Y_i=X_i+\epsilon_i$, $ i=1,...,n+1$ with $(X_i)$ a real-valued positive recurrent and stationary Markov chain and $(\epsilon_i)_{1\leq i\leq n+1}$ a noise independent of the sequence $(X_i)$ 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 $X_i$ and an estimator of the density of $(X_i,X_{i+1})$. 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.

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