Statistical inference with weak beliefs

Jianchun Zhang, Purdue University

Abstract

The Dempster-Shafer (DS) theory is a powerful tool for probabilistic reasoning based on a formal calculus for combing evidence. DS theory has been widely used in computer science and engineering applications, but has yet to reach the statistical mainstream, perhaps due to its computational difficulty, non-uniqueness, and lack of long-run frequency property. In this thesis, we return to Dempster's original approach to constructing belief functions for statistical inference, and propose to modify the DS models by systematically enlarging focal elements in order to obtain belief functions that have desired frequency properties. We call this the weak belief (WB) approach. We present a general description of WB by incorporating it into the framework of inferential models (IM), its interplay with the DS calculus, and the maximal belief (MB) solution. An essential task of WB inference is to propose efficient predictive random set (PRS) in light of the targeted inferential problems. We propose a couple of efficient PRSs for predicting the ordered uniform realizations. These PRSs, together with the MB solution, have wide applications in a series of statistical problems. Several applications are considered, such as inference about a binomial proportion, one-sample goodness-of-fit test, and large-scale hypothesis testing with its application to a real microarray data. The WB approach provides a rigorous way of solving statistical problems. Empirical results show that the WB approach performs well.

Degree

Ph.D.

Advisors

Liu, Purdue University.

Subject Area

Statistics

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