#### Date of Completion

7-22-2020

#### Embargo Period

7-19-2021

#### Keywords

Sequential Analysis, Purely Sequential Sampling, Asymptotic Efficiency, Asymptotic Consistency

#### Major Advisor

Nitis Mukhopadhyay

#### Associate Advisor

Joseph Glaz

#### Associate Advisor

Haim Bar

#### Field of Study

Statistics

#### Degree

Doctor of Philosophy

#### Open Access

Open Access

#### Abstract

Purely sequential procedure has been widely studied in different inference problems. However, in purely sequential procedure, only one observation should be taken at-a-time. In real life, packaged items purchased in bulk often cost less per unit sample than the cost of an individual item. This dissertation discussed this situation when observations are gathered in groups. First, two fundamental problems on purely sequential estimation are revisited: (i) the fixed-width confidence interval (FWCI) estimation problem, and (ii) the minimum risk point estimation (MRPE) problem, in the context of estimating an unknown mean in a normal population having an unknown variance. We begin by laying down general frameworks for the second-order asymptotic analyses, in both problems, under sequential sampling of one observation at-a-time. Then, we consider sequentially sampling k observations at-a-time in defining our proposed estimation strategies. In the first attempt, tentative estimators are used to study the feasibility. Then, replace the simple class of estimators with more complicated unbiased and consistent estimators under permutations within each group. These new estimators incorporated in the definition of the stopping boundaries have led to tighter estimation of requisite optimal fixed-sample sizes. In both scenarios, first-order and second-order asymptotic properties have been analyzed under appropriate requirements on the pilot sample size.

Such estimators can also be used in two-sample comparisons. The last part of this dissertation presents the second-order asymptotic properties for comparing treatment means. Two separate situations are considered: (i) σ_{1} = σ_{2} = σ, but σ is assumed unknown, and (ii) σ_{1 } and σ_{2} are unequal and unknown. For datasets with possible outliers, robust estimators are in use in purely sequential estimation strategies. For each problem, large-scale computer simulations and substantial data analysis have validated corresponding results. The methodologies are illustrated with the help of real-world data.

#### Recommended Citation

Wang, Zhe, "Sequential Estimation Methodologies with Observations Gathered in Groups: Theory, Practice and Data Analysis" (2020). *Doctoral Dissertations*. 2551.

https://opencommons.uconn.edu/dissertations/2551