Date of Completion

7-20-2018

Embargo Period

7-19-2020

Keywords

Fixed-width confidence interval, Gini's mean difference, mean absolute deviation, minimum risk point estimation, normal mean, purely sequential methodologies, two-stage methodologies

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

This thesis consists of multi-stage methodologies handling two fundamental estimation problems. These are (i) fixed-width confidence interval estimation (FCIE), and (ii) minimum risk point estimation (MRPE) problems for the unknown mean μ of a normal distribution whose variance σ² is also assumed unknown.

We first develop purely sequential estimation methodologies for both FCIE and MRPE problems. New stopping rules are constructed by replacing the sample variance with appropriate multiples of Gini's mean difference (GMD) and mean absolute deviation (MAD) in defining the conditions for boundary crossing. A number of asymptotic first-order consistency, efficiency, and risk efficiency properties associated with these new estimation strategies has been investigated. These are followed by summaries obtained from extensive sets of simulations by drawing samples from (i) normal universes or (ii) mixture-normal universes where samples may be reasonably treated as observations from a normal universe in a large majority of simulations. We also include illustrations using sales data and horticulture data.

By revisiting Stein (1945,1949) as well as Mukhopadhyay and Duggan (1997), we then move on to propose new two-stage estimation methodologies under both MRPE and FCIE configurations for a normal mean μ when a lower bound of variance σL2 (0<σL

Next, we further explore the asymptotic second-order approximations for the regret function associated with the purely sequential MRPE methodologies, providing a general structure.

Overall, we empirically feel confident that our newly developed GMD-based or MAD-based multi-stage estimation methodologies are more robust for practical purposes when we compare them with the sample variance-based methodologies respectively, especially when suspect outliers may be expected. We conclude with some interesting directions of future research work that we can follow to make our multi-stage estimation methodologies more widely applicable for a lot of inference problems.

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