Title

On some bounded risk sequential procedures for exponential mean and normal density estimation

Date of Completion

January 2007

Keywords

Statistics

Degree

Ph.D.

Abstract

This dissertation will look at three problems in the area of sequential analysis. The first problem we discuss is point estimation of the mean of an exponential distribution following a two-stage sampling design. We assume the associated risk is bounded by some positive pre-assigned risk bound. The goal here is to develop a genuine two-stage sampling procedure in the sense of Stein (1945,1949) where an exact risk bound has been achieved. We note that the terminal sample size and estimator are dependent. The performances of the proposed methodology are investigated with the help of simulations. Illustrations are also included with the help of real data from a multicenter clinical trial. ^ The second problem we discuss is plug-in density estimation of a normal distribution with zero mean and unknown variance via two-stage and purely sequential sampling. We propose to estimate the normal density function under consideration by using the mean integrated squared error loss function. Our goal is to make the associated risk not to exceed a preassigned positive number. Since no fixed-sample-size methodology would be able to handle this estimation problem, we design appropriate two-stage and purely sequential density estimation methodologies that are both shown to satisfy the asymptotic first-order efficiency property, the first-order risk-efficiency property, as well as the second-order efficiency property. Comparisons between the two methodologies are discussed. The performances of the proposed methodologies are investigated with the help of simulations. Robustness of the proposed methodologies under mixture-normal population densities are considered. Illustrations are included with the help of real data and analysis. ^ The third problem we look at is very similar to the second problem. We discuss plug-in density estimation of a normal distribution with unknown mean and unknown variance via two-stage and purely sequential sampling. Our goal remains the same as in the second problem. Likewise, we show the methodologies satisfy the properties above. Comparisons between the two methodologies are discussed, simulations are provided, robustness considerations are investigated, and real data illustrations are included. ^