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

2006/11/22 by Claire Lacour
Computer Science · Economics, Econometrics and Finance · Mathematics · #Adaptive estimator #Applied mathematics #Artificial intelligence #Bayesian Methods and Mixture Models #Computer science #Density estimation #Estimator #Financial Risk and Volatility Modeling #Hidden Markov model #Markov chain #Markov chain mixing time #Markov model #Mathematics #Sequence (biology) #Stationary distribution #Statistical Methods and Inference #Statistics #Variable-order Markov model #math.ST #stat.TH

paper · pdf · doi:10.1016/j.jmva.2007.04.006

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

arxiv created 2006/11/22 · openalex publication_date 2007/04/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We study the following model of hidden Markov chain: Y_i=X_i+ε_i, i=1,...,n+1 with (X_i) a real-valued positive recurrent and stationary Markov chain and (ε_i)_{1≤ i≤ 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.

Citations

• Statistical Inference for Probabilistic Functions of Finite State Markov Chains
• Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime
• On the Optimal Rates of Convergence for Nonparametric Deconvolution Problems
• Concentration around the mean for maxima of empirical processes