Nonparametric estimation of the stationary density and the transition density of a Markov chain

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Abstract

In this paper, we study first the problem of nonparametric estimation of the stationary density f of a discrete-time Markov chain (Xi). We consider a collection of projection estimators on finite dimensional linear spaces. We select an estimator among the collection by minimizing a penalized contrast. The same technique enables us to estimate the density g of (Xi,Xi+1) and so to provide an adaptive estimator of the transition density π=g/f. We give bounds in L2 norm for these estimators and we show that they are adaptive in the minimax sense over a large class of Besov spaces. Some examples and simulations are also provided.

Keywords

Adaptive estimation
Markov chain
Stationary density
Transition density
Model selection
Penalized contrast
Projection estimators

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