Shape constrained density estimation

• Author / Creator

  Lin, Mu

• 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.

• Subjects / Keywords

  • Classic Guassian compound decision problem
  • Rho-concave density estimation
  • Empirical Bayes inference
  • Rho-consistency, Hellinger consistency, total variation consistency
  • Shape constrained density estimation

• Graduation date

  Spring 2014

• Type of Item

  Thesis

• Degree

  Doctor of Philosophy

• DOI

  https://doi.org/10.7939/R3125QJ2K

• License

  This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.