Date of Completion

7-17-2017

Embargo Period

7-17-2017

Keywords

Statistics, Sequential Estimation

Major Advisor

Nitis Mukhopadhyay

Associate Advisor

Joseph Glaz

Associate Advisor

Vladimir Pozdnyakov

Field of Study

Statistics

Degree

Doctor of Philosophy

Open Access

Open Access

Abstract

A sequential sampling methodology provides concrete results and proves to be benefecial in many scenarios, where a fixed sampling technique fails to deliver. This dissertation introduces several multistage sampling methodologies to estimate the unknown parameters depending on the model in hand. We construct both two-stage and purely sequential sampling rules under different situations. The estimation is carried under a loss function which in our case is either a usual squared error loss or a Linex loss. We adopt a technique known as bounded risk estimation strategy, where we bound the appropriate risk function from above by a fixed and known constant. At first we draw attention to a negative exponential distribution and applications from health studies. We propose appropriate stopping rules to estimate the location parameter or the threshold of a negative exponential distribution under a Linex loss function. This model proves to be relevant to depict failure times of complex equipment or survival times in cancer research. We include some real data applications such as to estimate the minimum threshold of infant mortality rates for different countries.

We then move on to extend this investigation to a two-sample situation, where we estimate the difference in locations of two independent negative exponential populations. The estimation is again carried out under Linex loss. We introduce some applications from cancer studies and reliability analysis.

The third fold of this dissertation involves the bounded risk multistage point estimation of a negative binomial (NB) mean under different loss functions. We assume that the thatch parameter is either known or unknown. We use a parameterization of the NB model which was first introduced by Anscombe in (1949). This is on slightly different lines since it involves a discrete population. A negative binomial model finds its use in entomological or ecological studies involving count data. We propose two-stage and purely sequential rules under squared error and Linex loss functions. We include real data applications involving weed count and bird count data. We next move on to extend this work for a multi-sample situation where we 1) simultaneously estimate a k-vector of NB means and 2) estimate the difference in means of two independent NB populations. We again assume that the thatch parameters are either known or unknown. In the case when the thatch parameters are unknown, we have designed an interesting allocation scheme along with suitable set of stopping rules. The work is supported using interesting real world applications. We should mention that all of our proposed methodologies enjoy exciting asymptotic efficiency and consistency properties depending on the scenario. Finally, we conclude by discussing some attractive areas of future research that may be of practical significance.

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