Abstract. The aim of this paper is to present a new estimation procedure that can be applied in various statistical frameworks including density and regression and which leads to both robust and optimal (or nearly optimal) estimators. In density estimation, they asymptotically coincide with the celebrated maximum likelihood estimators at least when the statistical model is regular enough and contains the true density to estimate. For very general models of densities, including non-compact ones, these estimators are robust with respect to the Hellinger distance and converge at optimal rate (up to a possible logarithmic factor) in all cases we know. In the regression setting, our approach improves upon the classical least squares in many respects. In simple linear regression for example, it provides an estimation of the coefficients that are both robust to outliers and simultaneously rateoptimal (or nearly rate-optimal) for a large class of error distributions including Gaussian, Laplace, Cauchy and uniform among others.

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

Abstract. We propose a new estimation procedure of the conditional density for independent and identically distributed data. Our procedure aims at using the data to select a function among arbitrary (at most countable) collections of candidates. By using a deterministic Hellinger distance as loss, we prove that the selected function satisfies a non-asymptotic oracle type inequality under minimal assumptions on the statistical setting. We derive an adaptive piecewise constant estimator on a random partition that achieves the expected rate of convergence over (possibly inhomogeneous and anisotropic) Besov spaces of small regularity. Moreover, we show that this oracle inequality may lead to a general model selection theorem under very mild assumptions on the statistical setting. This theorem guarantees the existence of estimators possessing nice statistical properties under various assumptions on the conditional density (such as smoothness or structural ones).Mathematics Subject Classification. 62G05, 62G07.