In this paper we are interested in the estimation of a density − defined on a compact interval ofℝ− from n independent andidentically distributed observations. In order to avoid boundary effect, beta kernelestimators are used and we propose a procedure (inspired by Lepski’s method) in order toselect the bandwidth. Our procedure is proved to be adaptive in an asymptotically minimaxframework. Our estimator is compared with both the cross-validation algorithm and theoracle estimator using simulated data.

Let X be a one dimensional positive recurrent diffusion continuously observed on [0,t] . We consider a non parametric estimator of the drift function on a given interval. Our estimator, obtained using a penalized least square approach, belongs to a finite dimensional functional space, whose dimension is selected according to the data. The non-asymptotic risk-bound reaches the minimax optimal rate of convergence when t → ∞. The main point of our work is that we do not suppose the process to be in stationary regime neither to be exponentially β-mixing. This is possible thanks to the use of a new polynomial inequality in the ergodic theorem [E. Löcherbach, D. Loukianova and O. Loukianov, Ann. Inst. H. Poincaré Probab. Statist. 47 (2011) 425–449].

Assume that (Xt )t∈Z is a real valued time seriesadmitting a common marginal density f with respect to Lebesgue's measure. [Donoho et al. Ann. Stat.24 (1996) 508–539] propose near-minimax estimators $\widehat f_n$ based on thresholding wavelets to estimate f on a compact set in an independent and identically distributed setting. The aim of the present work is to extend these results to general weak dependent contexts. Weak dependence assumptions are expressed as decreasing bounds of covariance terms and are detailed for different examples. The threshold levels in estimators $\widehat f_n$ depend on weak dependence properties of the sequence (Xt )t∈Z through the constant. If these properties are unknown, we propose cross-validation procedures to get new estimators. These procedures are illustrated via simulations of dynamical systems and non causal infinite moving averages. We also discuss the efficiency of our estimators with respect to the decrease of covariances bounds.

A method is introduced to select the significant or non null mean terms among a collection of independent random variables. As an application we consider the problem of recovering thesignificant coefficients in non ordered model selection. The method is based on a convenient random centering ofthe partial sums of the ordered observations. Based onL-statistics methods we show consistency of the proposedestimator.An extension to unknown parametric distributions is considered. Simulatedexamples are included to show the accuracy of the estimator.An example of signal denoising with wavelet thresholding is also discussed.

We want to recover a signal based on noisy inhomogeneous data (the amount of data can vary strongly on the estimation domain). We model the data using nonparametric regression with random design, and we focus on the estimation of the regression at a fixed point x 0 with little, or much data. We propose a method which adapts both to the local amount of data (the design density is unknown) and to the local smoothness of the regression function. The procedure consists of a local polynomial estimator with a Lepski type data-driven bandwidth selector, see for instance Lepski et al. [Ann. Statist.25 (1997) 929–947]. We assess this procedure in the minimax setup, over a class of function with local smoothness s >0 of Hölder type. We quantify the amount of data at x 0 in terms of a local property on the design density called regular variation, which allows situations with strong variations in the concentration of the observations. Moreover, the optimality of the procedure is proved within this framework.

We consider the problem of estimating the integral of the square of a densityf from the observation of a n sample. Our method to estimate $\int_{\mathbb{R}} f^2(x){\rm d}x$ isbased on model selection via some penalized criterion. We prove that our estimator achieves the adaptive rates established by Efroimovich and Low on classes of smooth functions. A key point of the proof is an exponentialinequality for U-statistics of order 2 due to Houdré and Reynaud.

An adaptive Kalman filtering approach is proposed for attitude determination to replace the fixed (conventional) Kalman filtering approach. The filter is used to adaptively reflect system dynamics changes or rapid changes in vehicle trajectory. The estimation procedure is carried out through the use of a measurement residual sequence. The sequence is used as a piece-wise stationary process inside an estimation window. The measurement noise covariance matrix and the system noise matrix are adaptively estimated. An extended Kalman filter approach with iteration of the states within-an-epoch was performed to overcome the non-linearity of the observation equations. A test was performed to evaluate the proposed technique. Different trajectory scenarios are presented to show the difference in performance between the adaptive Kalman filter and the conventional one. Results show that the proposed adoptive filtering approach has a better performance than the conventional filter.

In this paper a brief review of Sage adaptive filtering is followed by an analysis of the shortcomings of covariance matrices formed by windowing residual vectors, innovation vectors and correction vectors of the dynamic states. A new adaptive Kalman filter is developed by combining the Sage filter and the variance components and its use tested against various other schemes.

In this paper, we study the problem of non parametric estimationof the stationary marginal density f of an α or aβ-mixing process, observed either in continuous time or indiscrete time. We present an unified framework allowing to dealwith many different cases. We consider a collection of finitedimensional linear regular spaces. We estimate f using aprojection estimator built on a data driven selected linear spaceamong the collection. This data driven choice is performed via theminimization of a penalized contrast. We state non asymptotic riskbounds, regarding to the integrated quadratic risk, for ourestimators, in both cases of mixing. We show that they areadaptive in the minimax sense over a large class of Besov balls.In discrete time, we also provide a result for model selectionamong an exponentially large collection of models (non regularcase).

In this paper, we study the problem of non parametric estimationof an unknown regression function from dependent data withsub-Gaussian errors. As a particular case, we handle theautoregressive framework. For this purpose, we consider acollection of finite dimensional linear spaces (e.g. linear spacesspanned by wavelets or piecewise polynomials on a possiblyirregular grid) and we estimate the regression function by aleast-squares estimator built on a data driven selected linearspace among the collection. This data driven choice is performedvia the minimization of a penalized criterion akin to the Mallows'C p . We state non asymptotic risk bounds for our estimator insome ${\mathbb{L}}_2$ -norm and we show that it is adaptive in the minimaxsense over a large class of Besov balls of the form Bα,p,∞(R) with p ≥ 1.