Estimation of the transition density of a Markov chain

Abstract

We present two data-driven procedures to estimate the transition density of an homogeneous Markov chain. The first yields to a piecewise constant estimator on a suitable random partition. By using an Hellinger-type loss, we establish non-asymptotic risk bounds for our estimator when the square root of the transition density belongs to possibly inhomogeneous Besov spaces with possibly small regularity index. Some simulations are also provided. The second procedure is of theoretical interest and leads to a general model selection theorem from which we derive rates of convergence over a very wide range of possibly inhomogeneous and anisotropic Besov spaces. We also investigate the rates that can be achieved under structural assumptions on the transition density. Introduction.Consider a time-homogeneous Markov chain (X i ) i∈N defined on an abstract probability space (Ω, E, P ) with values in the measured space (X, F, µ). We assume that for each x ∈ X, the conditional law L(X i+1 | X i = x) admits a density s(x, ·) with respect to µ. Our aim is to estimate the transition density (x, y) → s(x, y) on a subset A = A 1 × A 2 of X 2 from the observations X 0 , . . . , X n .Many papers are devoted to this statistical setting. A popular method to build an estimator of s is to divide an estimator of the joint density of (X i , X i+1 ) by an estimator of the density of X i . The resulting estimator is called a quotient estimator. Roussas (1969), Athreya and Atuncar (1998) considered Kernel estimators for the densities of X i and (X i , X i+1 ). They proved consistence and asymptotic normality of the quotient estimator. Other properties of this estimator were established: Roussas (1991), Dorea (2002) showed strong consistency, Basu and Sahoo (1998) proved a Berry-Essen type theorem and Doukhan and Ghindès (1983) bounded from above the integrated quadratic risk under Sobolev constraints. Clémencon (2000) investigated the minimax rates when A = [0, 1] 2 , X 2 = R 2 . Given two smoothness classes F 1 and F 2 of real valued functions on [0, 1] 2 and [0, 1] respectively (balls of Besov spaces), he established the lower bounds over the classHe developed a method based on wavelet thresholding to estimate the densities of X i and (X i , X i+1 ) and showed that the quotient estimator of s is quasi-optimal in the sense that the

Estimating the conditional density by histogram type estimators and model selection

“…The notation · ∞ stands for the supremum norm: g ∞ = sup x∈[0,1] |g(x)|. An upper bound for ε F (f ) may be found in Section 4.4 of Sart (2014). Actually, we show in Section 4.6 that this bound can be slightly improved.…”

“…The definition of the criterion γ looks like the one proposed in Section 4.1 of Sart (2014) for estimating the transition density of a Markov chain as well as the one proposed in Baraud et al (2016) for estimating one or several densities. The underlying idea is that γ(f ) + L∆(f )/n is roughly between h 2 (s, f ) and h 2 (s, f ) + L∆(f )/n.…”

“…The discretization trick described in Section 4.2 of Sart (2014) can be adapted to our statistical setting. It leads to the theorem below.…”

“…The flexibility of our approach enables the study of various assumptions as illustrated by the three examples below. We refer to Sart (2014); Baraud and Birgé (2014) for additional examples. In the remainder of this section, µ and ν stand for the Lebesgue measure.…”

“…They show how to build collections V of linear spaces V with good approximation properties with respect to composite functions f of the form f = g • u. Using these results is the strategy of Sart (2014) to get bounds on ε F (f ) for classes F corresponding to structural assumptions on s. Nevertheless, this direct application of the results of Baraud and Birgé (2014) (with τ = log n/n) leads to an unnecessary additional logarithmic term in the risk bounds. A careful look at the proof of Theorem 2 of Baraud and Birgé (2014) shows that the following result holds.…”