Date of Completion

8-15-2017

Embargo Period

8-15-2018

Major Advisor

Ming-Hui Chen

Co-Major Advisor

Lynn Kuo

Associate Advisor

Paul O. Lewis

Field of Study

Statistics

Degree

Doctor of Philosophy

Open Access

Open Access

Abstract

Various modifications have been suggested in the past to extend the Shannon entropy to continuous random variables. We propose a new entropy called the fractional size adjusted entropy and later extend it to the generalized fractional size adjusted entropy. These two proposed entropies always exist and maintain non-negative values. The generalized fractional size adjusted entropy also includes many well-known entropies as its special cases, such as the Shannon entropy and the Renyi entropy. We apply our proposed entropies on various distributions and a phylogenetic example to demonstrate their good performances.

In addition, we propose a partition based measure to quantify the compatibility of two data sets using their respective posterior distributions. It is of great practical importance to compare and combine data from different studies in order to carry out appropriate and more powerful statistical inference. We further propose an information gain measure to quantify the information increase (or decrease) in combining two data sets. The compatibility measure and the information gain measure are well calibrated and efficient computational algorithms are provided for their calculations. We use a benchmark toxicology example, a six cities longitudinal health study and a melanoma clinical trials to illustrate how these measures are useful in combining current data with historical data and missing data.

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