Concentration inequalities are used to derive some new inequalities for ratio-type suprema of empirical processes. These general inequalities are used to prove several new limit theorems for ratio-type suprema and to recover anumber of the results from [1] and [2]. As a statistical application, an oracle inequality for nonparametric regression is obtained via ratio bounds. 1.

We consider a sequence space model of statistical linear inverse problems where we need to estimate a function f from indirect noisy observations. Let a finite set of linear estimators be given. Our aim is to mimic the estimator in that has the smallest risk on the true f . Under general conditions, we show that this can be achieved by simple minimization of unbiased risk estimator, provided the singular values of the operator of the inverse problem decrease as a power law. The main result is a nonasymptotic oracle inequality that is shown to be asymptotically exact. This inequality can be also used to obtain sharp minimax adaptive results. In particular, we apply it to show that minimax adaptation on ellipsoids in multivariate anisotropic case is realized by minimization of unbiased risk estimator without any loss of efficiency with respect to optimal non-adaptive procedures. Mathematics Subject Classifications: 62G05, 62G20 Key Words: Statistical inverse problems, Oracle inequaliti...

An unknown signal plus white noise is observed at n discrete time points. Within a large convex class of linear estimators of , we choose the estimator b that minimizes estimated quadratic risk. By construction, b is nonlinear. This estimation is done after orthogonal transformation of the data to a reasonable coordinate system. The procedure adaptively tapers the coefficients of the transformed data. If the class of candidate estimators satisfies a uniform entropy condition, then b is asymptotically minimax in Pinsker's sense over certain ellipsoids in the parameter space and shares one such asymptotic minimax property with the James-Stein estimator. We describe computational algorithms for b and construct confidence sets for the unknown signal. These confidence sets are centered at b , have correct asymptotic coverage probability, and have relatively small risk as set-valued estimators of .

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This work deals with a generalization of the Total Least Squares method in the context of the functional linear model. We first propose a smoothing splines estimator of the functional coefficient of the model without noise in the covariates and we obtain an asymptotic result for this estimator. Then, we adapt this estimator to the case where the covariates are noisy and we also derive an upper bound for the convergence speed. Our estimation procedure is evaluated by means of simulations.

Abstract. This paper studies statistical aggregation procedures in regression setting. A motivating factor is the existence of many different methods of estimation, leading to possibly competing estimators. We consider here three different types of aggregation: model selection (MS) aggregation, convex (C) aggregation and linear (L) aggregation. The objective of (MS) is to select the optimal single estimator from the list; that of (C) is to select the optimal convex combination of the given estimators; and that of (L) is to select the optimal linear combination of the given estimators. We are interested in evaluating the rates of convergence of the excess risks of the estimators obtained by these procedures. Our approach is motivated by recent minimax results in Nemirovski (2000) and Tsybakov (2003). There exist competing aggregation procedures achieving optimal convergence separately for each one of (MS), (C) and (L) cases. Since the bounds in these results are not directly comparable with each other, we suggest an alternative solution. We prove that all the three optimal bounds can be nearly achieved via a single “universal ” aggregation procedure. We propose such a procedure which consists in mixing of the initial estimators with the weights obtained by penalized least squares. Two different penalities are considered: one of them is related to hard thresholding techniques, the second one is a data dependent L1-type penalty. 1.

The method of regularization with the Gaussian reproducing kernel is popular in the machine learning literature and successful in many practical applications. In this paper we consider the periodic version of the Gaussian kernel regularization. We show in the white noise model setting, that in function spaces of very smooth functions, such as the infinite-order Sobolev space and the space of analytic functions, the method under consideration is asymptotically minimax; in finite-order Sobolev spaces, the method is rate optimal, and the efficiency in terms of constant when compared with the minimax estimator is reasonably high. The smoothing parameters in the periodic Gaussian regularization can be chosen adaptively without loss of asymptotic efficiency. The results derived in this paper give a partial explanation of the success of the Gaussian reproducing kernel in practice. Simulations are carried out to study the finite sample properties of the periodic Gaussian regularization. 1. Introduction. The

The question of recovering a multiband signal from noisy observations motivates a model in which the multivariate data points consist of an unknown deterministic trend # observed with multivariate Gaussian errors. A cognate random trend model suggests affine shrinkage estimators #A and #B for #, which are related to an extended Efron-Morris estimator. When represented canonically, #A performs componentwise James-Stein shrinkage in a coordinate system that is determined by the data. Under the original deterministic trend model, #A and its relatives are asymptotically minimax in Pinsker's sense over certain classes of subsets of the parameter space. In such fashion, #A and its cousins dominate the classically efficient least squares estimator. We illustrate their use to improve on the least squares fit of the multivariate linear model.

Non-parametric estimation in a

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