Keywords

Empirical bayes, functional deconvolution, minimax convergence rate, wavelets

Abstract

In the present dissertation, we investigate two different nonparametric models; empirical Bayes model and functional deconvolution model. In the case of the nonparametric empirical Bayes estimation, we carried out a complete minimax study. In particular, we derive minimax lower bounds for the risk of the nonparametric empirical Bayes estimator for a general conditional distribution. This result has never been obtained previously. In order to attain optimal convergence rates, we use a wavelet series based empirical Bayes estimator constructed in Pensky and Alotaibi (2005). We propose an adaptive version of this estimator using Lepski’s method and show that the estimator attains optimal convergence rates. The theory is supplemented by numerous examples. Our study of the functional deconvolution model expands results of Pensky and Sapatinas (2009, 2010, 2011) to the case of estimating an (r + 1)-dimensional function or dependent errors. In both cases, we derive minimax lower bounds for the integrated square risk over a wide set of Besov balls and construct adaptive wavelet estimators that attain those optimal convergence rates. In particular, in the case of estimating a periodic (r + 1)-dimensional function, we show that by choosing Besov balls of mixed smoothness, we can avoid the ”curse of dimensionality” and, hence, obtain higher than usual convergence rates when r is large. The study of deconvolution of a multivariate function is motivated by seismic inversion which can be reduced to solution of noisy two-dimensional convolution equations that allow to draw inference on underground layer structures along the chosen profiles. The common practice in seismology is to recover layer structures separately for each profile and then to combine the derived estimates into a two-dimensional function. By studying the two-dimensional version of the model, we demonstrate that this strategy usually leads to estimators which are less accurate than the ones obtained as two-dimensional functional deconvolutions. Finally, we consider a multichannel deconvolution model with long-range dependent Gaussian errors. We do not limit our consideration to a specific type of long-range dependence, rather we assume that the eigenvalues of the covariance matrix of the errors are bounded above and below. We show that convergence rates of the estimators depend on a balance between the smoothness parameters of the response function, the iii smoothness of the blurring function, the long memory parameters of the errors, and how the total number of observations is distributed among the channels.

Notes

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Graduation Date

2013

Semester

Summer

Advisor

Pensky, Marianna

Degree

Doctor of Philosophy (Ph.D.)

College

College of Sciences

Department

Mathematics

Degree Program

Mathematics

Format

application/pdf

Identifier

CFE0004814

URL

http://purl.fcla.edu/fcla/etd/CFE0004814

Language

English

Release Date

August 2013

Length of Campus-only Access

None

Access Status

Doctoral Dissertation (Open Access)

Subjects

Dissertations, Academic -- Sciences, Sciences -- Dissertations, Academic

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Included in

Mathematics Commons

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