Least squares type estimation of the transition density of a particular hidden Markov chain

In this paper, 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 stationary Markov chain and $(\epsilon_i)_{1\leq i\leq n+1}$ a noise having a known distribution and independent of the sequence $(X_i)$. We present an estimator of the transition density obtained by minimization of an original contrast that takes advantage of the regressive aspect of the problem. It is selected among a collection of projection estimators with a model selection method. The $L^2$-risk and its rate of convergence are evaluated for ordinary smooth noise and some simulations illustrate the method. We obtain uniform risk bounds over classes of Besov balls. In addition our estimation procedure requires no prior knowledge of the regularity of the true transition. Finally, our estimator permits to avoid the drawbacks of quotient estimators.Comment: Published in at http://dx.doi.org/10.1214/07-EJS111 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

Rates of convergence for nonparametric deconvolution

This Note presents original rates of convergence for the deconvolution problem. We assume that both the estimated density and noise density are supersmooth and we compute the risk for two kinds of estimators

Adaptive estimation of the transition density of a Markov chain

In this paper a new estimator for the transition density $\pi$ of an homogeneous Markov chain is considered. We introduce an original contrast derived from regression framework and we use a model selection method to estimate $\pi$ under mild conditions. The resulting estimate is adaptive with an optimal rate of convergence over a large range of anisotropic Besov spaces $B_{2,\infty}^{(\alpha_1,\alpha_2)}$. Some simulations are also presented

Minimal penalty for Goldenshluger-Lepski method

This paper is concerned with adaptive nonparametric estimation using the Goldenshluger-Lepski selection method. This estimator selection method is based on pairwise comparisons between estimators with respect to some loss function. The method also involves a penalty term that typically needs to be large enough in order that the method works (in the sense that one can prove some oracle type inequality for the selected estimator). In the case of density estimation with kernel estimators and a quadratic loss, we show that the procedure fails if the penalty term is chosen smaller than some critical value for the penalty: the minimal penalty. More precisely we show that the quadratic risk of the selected estimator explodes when the penalty is below this critical value while it stays under control when the penalty is above this critical value. This kind of phase transition phenomenon for penalty calibration has already been observed and proved for penalized model selection methods in various contexts but appears here for the first time for the Goldenshluger-Lepski pairwise comparison method. Some simulations illustrate the theoretical results and lead to some hints on how to use the theory to calibrate the method in practice

Estimator selection: a new method with applications to kernel density estimation

Estimator selection has become a crucial issue in non parametric estimation. Two widely used methods are penalized empirical risk minimization (such as penalized log-likelihood estimation) or pairwise comparison (such as Lepski's method). Our aim in this paper is twofold. First we explain some general ideas about the calibration issue of estimator selection methods. We review some known results, putting the emphasis on the concept of minimal penalty which is helpful to design data-driven selection criteria. Secondly we present a new method for bandwidth selection within the framework of kernel density density estimation which is in some sense intermediate between these two main methods mentioned above. We provide some theoretical results which lead to some fully data-driven selection strategy

Numerical performance of Penalized Comparison to Overfitting for multivariate kernel density estimation

Kernel density estimation is a well known method involving a smoothing parameter (the bandwidth) that needs to be tuned by the user. Although this method has been widely used the bandwidth selection remains a challenging issue in terms of balancing algorithmic performance and statistical relevance. The purpose of this paper is to compare a recently developped bandwidth selection method for kernel density estimation to those which are commonly used by now (at least those which are implemented in the R-package). This new method is called Penalized Comparison to Overfitting (PCO). It has been proposed by some of the authors of this paper in a previous work devoted to its statistical relevance from a purely theoretical perspective. It is compared here to other usual bandwidth selection methods for univariate and also multivariate kernel density estimation on the basis of intensive simulation studies. In particular, cross-validation and plug-in criteria are numerically investigated and compared to PCO. The take home message is that PCO can outperform the classical methods without algorithmic additionnal cost

Adaptive pointwise estimation for pure jump L\'evy processes

This paper is concerned with adaptive kernel estimation of the L\'evy density N(x) for bounded-variation pure-jump L\'evy processes. The sample path is observed at n discrete instants in the "high frequency" context (\Delta = \Delta(n) tends to zero while n\Delta tends to infinity). We construct a collection of kernel estimators of the function g(x)=xN(x) and propose a method of local adaptive selection of the bandwidth. We provide an oracle inequality and a rate of convergence for the quadratic pointwise risk. This rate is proved to be the optimal minimax rate. We give examples and simulation results for processes fitting in our framework. We also consider the case of irregular sampling

Adaptive estimation of the dynamics of a discrete time stochastic volatility model

International audienceThis paper is concerned with the particular hidden model: $X_{i+1}=b(X_i)+\sigma(X_i)\xi_{i+1}, Z_i=X_i+\varepsilon_i$, where $(\xi_i)$ and $(\varepsilon_i)$ are independent sequences of i.i.d. noise. Moreover, the sequences $(X_i)$ and $(\varepsilon_i)$ are independent and the distribution of $\varepsilon$ is known. Our aim is to estimate the functions $b$ and $\sigma^2$ when only observations $Z_1, \dots, Z_n$ are available. We propose to estimate $bf$ and $(b^2+\sigma^2)f$ and study the integrated mean square error of projection estimators of these functions on automatically selected projection spaces. By ratio strategy, estimators of $b$ and $\sigma^2$ are then deduced. The mean square risk of the resulting estimators are studied and their rates are discussed. Lastly, simulation experiments are provided: constants in the penalty functions defining the estimators are calibrated and the quality of the estimators is checked on several examples

Minimax estimation of the conditional cumulative distribution function under random censorship

International audienceConsider an i.i.d. sample $(X_i,Y_i)$, $i=1, \dots, n$ of observations and denote by $F(x,y)$ the conditional cumulative distribution function of $Y_i$ given $X_i=x$. We provide a data driven nonparametric strategy to estimate $F$. We prove that, in term of the integrated mean square risk on a compact set, our estimator performs a squared-bias variance compromise. We deduce from this an upper bound for the rate of convergence of the estimator, in a context of anisotropic function classes. A lower bound for this rate is also proved, which implies the optimality of our estimator. Then our procedure can be adapted to positive censored random variables $Y_i$'s, i.e. when only $Z_i=\inf(Y_i, C_i)$ and \delta_i=\1_{\{Y_i\leq C_i\}} are observed, for an i.i.d. censoring sequence $(C_i)_{1\leq i\leq n}$ independent of $(X_i,Y_i)_{1\leq i\leq n}$. Simulation experiments illustrate the method