Alexander Meister

Deconvolution problems in nonparametric statistics

Density Deconvolution.- Nonparametric Regression with Errors-in-Variables.- Image and Signal Reconstruction.

A ridge-parameter approach to deconvolution

Kernel methods for deconvolution have attractive features, and prevail in the literature. However, they have disadvantages, which include the fact that they are usually suitable only for cases where the error distribution is infinitely supported and its characteristic function does not ever vanish. Even in these settings, optimal convergence rates are achieved by kernel estimators only when the kernel is chosen to adapt to the unknown smoothness of the target distribution. In this paper we suggest alternative ridge methods, not involving kernels in any way. We show that ridge methods (a) do not require the assumption that the error-distribution characteristic function is nonvanishing; (b) adapt themselves remarkably well to the smoothness of the target density, with the result that the degree of smoothness does not need to be directly estimated; and (c) give optimal convergence rates in a broad range of settings.

Density estimation with heteroscedastic error

veloped for homoscedastic errors become inconsistent. In this paper, we introduce a kernel estimator of a density in the case of heteroscedastic contamination. We establish consistency of the estimator and show that it achieves optimal rates of convergence under quite general conditions. We study the limits of appli cation of the procedure in some extreme situations, where we show that, in some cases, our estimator is consistent, even when the scaling parameter of the error is unbounded. We suggest a modified estimator for the problem where the distribution of the errors is unknown, but replicated observations are available. Finally, an adaptive procedure for selecting the smoothing parameter is proposed and its finite-sample prop erties are investigated on simulated examples.

Nonparametric Regression Estimation in the Heteroscedastic Errors-in-Variables Problem

In the classical errors-in-variables problem, the goal is to estimate a regression curve from data in which the explanatory variable is measured with error. In this context, nonparametric methods have been proposed that rely on the assumption that the measurement errors are identically distributed. Although there are many situations in which this assumption is too restrictive, nonparametric estimators in the more realistic setting of heteroscedastic errors have not been studied in the literature. We propose an estimator of the regression function in such a setting and show that it is optimal. We give estimators in cases in which the error distributions are unknown and replicated observations are available. Practical methods, including an adaptive bandwidth selector for the errors-in-variables regression problem, are suggested, and their finite-sample performance is illustrated through simulated and real data examples.

Deconvolution from Fourier-oscillating error densities under decay and smoothness restrictions

This paper is concerned with deconvolution from error or blurring densities whose Fourier transforms have isolated zeros and show oscillatory behaviour; unlike conventional approaches where the Fourier transform decays about monotonously. We introduce specific estimation procedures based on local polynomial approximation in the Fourier domain. Under combined moment and smoothness conditions, we are able to improve the convergence rates compared to existing methods in density deconvolution. The corresponding minimax theory is derived. In compactly supported models as in signal deconvolution and Berkson regression, nearly optimal rates are achieved under conditions which are significantly weaker than those assumed in earlier papers.

Asymptotic equivalence of functional linear regression and a white noise inverse problem

We consider the statistical experiment of functional linear regression (FLR). Furthermore, we introduce a white noise model where one observes an Ito process, which contains the covariance operator of the corresponding FLR model in its construction. We prove asymptotic equivalence of FLR and this white noise model in LeCams sense under known design distribution. Moreover, we show equivalence of FLR and an empirical version of the white noise model for finite sample sizes. As an application, we derive sharp minimax constants in the FLR model which are still valid in the case of unknown design distribution.

Nonparametric Regression Analysis for Group Testing Data

Group testing is a procedure used to reduce the cost and increase the speed of large screening studies in which infection or contamination of individuals is detected by a test carried out on a sample of, for example, blood, urine, or water. Instead of testing the sample of each individual, the method involves pooling samples of groups of several individuals and testing those pooled samples. We construct a nonparametric procedure for estimating the conditional probability of contamination given an explanatory variable when the observations are pooled data of this type. We investigate asymptotic theoretical properties of the estimator and establish its consistency. The procedure requires selecting an important smoothing parameter, and we suggest a way to do this automatically from the data. We illustrate the numerical performance of the method on some simulated examples and on data from the National Health and Nutrition Examination Survey. We discuss extensions of the procedure to cases where the test is imprecise and the covariates are observed inaccurately, and to the multivariate setting. Supplemental materials including proofs, R codes, and additional simulation results are available from the online JASA website.

Deconvolving compactly supported densities

This paper addresses the statistical problem of density deconvolution under the condition that the density to be estimated has compact support. We introduce a new estimation procedure, which establishes faster rates of convergence for smooth densities as compared to the optimal rates for smooth densities with unbounded support. This framework also allows us to relax the usual condition of known error density with non-vanishing Fourier transform, so that a nonparametric class of densities is valid; therefore, even the shape of the noise density need not be assumed. These results can also be generalized for fast decaying densities with unbounded support. We prove optimality of the rates in the underlying experiment and study the practical performance of our estimator by numerical simulations.

Adaptive function estimation in nonparametric regression with one-sided errors

We consider the model of nonregular nonparametric regression where smoothness constraints are imposed on the regression function

Support estimation via moment estimation in presence of noise

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