Year: 2014
Type: article
Abstract: The normal mean estimation problem has a central role in statistical estimation.The maximum likelihood estimator (MLE) is a traditional estimator for this problem.Under the squared error loss, the MLE is unbiased, minimax, best invariant, and admissible when the parameter's dimension is one or two.However, when the parameter's dimension is three or higher, it is inadmissible and can be uniformly dominated by other estimators.In the past half century, a great amount of research has been devoted to developing optimal estimators in high dimensional settings.The Bayesian approaches are highly appealing for this problem.However, selecting a prior becomes challenging when the parameter's dimension is high.Subjective elicitation of prior knowledge is almost infeasible.Therefore, we have to turn to some formal methods.In this dissertation, we develop a class of new prior distributions, namely, the adaptive inverted-Beta (AIB) priors, that lead to Bayesian estimators that often outperform many common estimators in the literature when the parameter is high dimensional.In the first part, Chapter 2, we focus on the situations where the observations are independent from a high dimensional normal distribution.We incorporate both global and local parameters in our AIB priors, so that different dimensions of the parameter can be shrunk by different factors.Most existing priors in the literature assume that data have a certain sparsity level.Instead, we utilize a hierarchical First, I would like to thank my advisor, Dr. Xinyi Xu, for inspiring this project and advising me with endless patience.She is so generous to share her brilliant and creative ideas with me and direct me when I was lost.I sincerely appreciate the time she spent thinking about the observations I found and checking every page of this thesis.Without her
Author: Di Cao
Institution: The Ohio State University
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W812172223