Degree

Doctor of Philosophy (PhD)

Department

Mathematics

Document Type

Dissertation

Abstract

The problems of interval estimation for quantiles of normal distributions in one- and multi-sample cases are considered. For the one-sample case, we show that the classical confidence interval (CI) based on the noncentral t (NCT) distribution and the one based on the uniformly minimum variance unbiased estimator of the population quantile are equivalent. We also propose a simple closed-form alternative CI for a quantile based on a normal approximation to the NCT distribution. In addition, fiducial distributions for population quantiles are developed using both the NCT distribution and its normal approximation. On the basis of fiducial distributions, we develop approximate closed-form CIs for the ratio or difference of two normal quantiles. The properties of the proposed and existing CIs are evaluated and compared using Monte Carlo simulation. Building on these results, the problem of comparing several normal populations in terms of quantiles is examined. We describe existing and new statistical tests for detecting differences among percentiles of multiple normal populations. Tests for comparing means when the variances are unknown and arbitrary (the Behrens -– Fisher problem) arise as a special case. An improved version of the likelihood ratio test (LRT) and a parametric bootstrap test are proposed. The performance of the proposed tests is evaluated in terms of error rates and power and compared with that of existing methods. In addition, we also consider simultaneous pairwise confidence intervals which can be used to identify the pairs of quantiles that are significantly different. Finally, we consider the problem of constructing confidence intervals for a common quantile of several normal populations using data from multiple independent studies. Exact CIs obtained by inverting combined tests are proposed, along with a closed-form approximate fiducial CI. These intervals are evaluated and compared with respect to accuracy and precision. R functions for computing all CIs are provided in a supplementary file. The proposed and available methods are illustrated using several practical examples including the data from colorectal cancer studies and latency data from anesthesia procedures.

Date

2-6-2026

DOI

https://proquest.com/docview/3347938649

First Committee Chair

Kalimuthu Krishnamoorthy

First Committee Member

Mo Li

Second Committee Member

Yongli Sang

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