## Theses and Dissertations--Statistics

2016

#### Degree Name

Doctor of Philosophy (PhD)

#### Document Type

Doctoral Dissertation

#### College

Arts and Sciences

Statistics

Dr. Mai Zhou

#### Abstract

Traditionally the inference on trimmed means, Lorenz Curves, and partial AUC (pAUC) under ROC curves have been done based on the asymptotic normality of the statistics. Based on the theory of empirical likelihood, in this dissertation we developed novel methods to do statistical inferences on trimmed means, Lorenz curves, and pAUC. A common characteristic among trimmed means, Lorenz curves, and pAUC is that their inferences are not based on the whole set of samples. Qin and Tsao (2002), Qin et al. (2013), and Qin et al. (2011) recently published their re- searches on the inferences of trimmed means, Lorenz curves, and pAUC based on empirical likelihood method, where they treated the cutting points in the samples fixed at the sample quantiles. They concluded that the limiting distributions of the empirical likelihood tests had scaled chi-square distributions under the null hypotheses. In our novel empirical likelihood methods, we treat the cutting points as the nuisance parameter(s). We conduct the inferences on trimmed means, Lorenz Curves, and pAUC in two steps. First, we make inferences on the parameter interested ( trimmed means, Lorenz curves, or pAUC) and the nuisance parameter(s) (the cutting point(s) in the samples) simultaneously. Then we profile out the nuisance parameter(s) from the test statistics. Under the null hypotheses, the limiting distributions of our empirical likelihood methods are chi-square. We innovate a computational algorithm ’ELseesaw’ to accomplish our empirical likelihood method for the inference on pAUC. Eventually, we contribute a R package to implement our empirical likelihood inferences on trimmed means, Lorenz curves, and pAUC. The R package we have developed can be downloaded free-of-charge on the internet at http://www.ms.uky.edu/~mai/EmpLik.html.

#### Digital Object Identifier (DOI)

https://doi.org/10.13023/ETD.2016.512

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