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Shape constrained density estimation Open Access


Other title
rho-concave density estimation
rho-consistency, Hellinger consistency, total variation consistency
classic Guassian compound decision problem
shape constrained density estimation
empirical Bayes inference
Type of item
Degree grantor
University of Alberta
Author or creator
Lin, Mu
Supervisor and department
Mizera, Ivan (Department of Mathematical and Statistical Sciences)
Examining committee member and department
Bowman, John (Department of Mathematical and Statistical Sciences)
Samworth, Richard (Department of Pure Mathematics and Mathematical Statistics)
Mizera, Ivan (Department of Mathematical and Statistical Sciences)
Prasad, Narasimha (Department of Mathematical and Statistical Sciences)
Karunamuni, Rohana (Department of Mathematical and Statistical Sciences)
Department of Mathematical and Statistical Sciences
Date accepted
Graduation date
Doctor of Philosophy
Degree level
We discuss non-parametric shape constrained density estimation methods in univariate setting and their applications to the classic Gaussian compound decision problem. The original contribution of the thesis is establishing various important consistency results of the shape constrained density estimators, which clarify the theoretical properties in the rho-concave density estimation problem and the mixture density estimation in classical Gaussian compound decision problem. Our main results begin with the consistency properties of rho-concave density estimator in quasi-concave density estimation problem, proposed by Koenker and Mizera (2010). We consider a new type of divergence called rho-divergence and prove the rho-consistency for the corresponding rho-concave density estimator when rho<0. We also generalize this consistency result to the consistencies under the Hellinger and the total variation distance. Next, we consider the monotone constrained mixture density estimation problem in the classical Gaussian compound decision problem. We first obtain the Hellinger consistency of the mixture density estimator and further adopt the similar formulation of the convex transformed maximum likelihood density estimation method of Seregin and Wellner (2010) to prove the pointwise consistency of the estimated convex function and decision rule in the interior of the domain of the true convex function. At last, we propose some new mixture density estimation approaches by imposing additional log-concave shape constraint on both the original monotone constrained maximum likelihood estimation and Kiefer-Wolfowitz maximum likelihood mixing distribution estimation methods respectively. Finally, we perform a simulation study to compare the new methods with various existing ones in the empirical Bayes inference problems.
Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
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