Theory and Methodologies of Multi-Stage Estimation for Comparing Location Parameters from Two Negative Exponential Populations and Illustrations with Data Analysis

Abstract

This thesis consists of multi-stage methodologies handling two fundamental estimation problems. These are (i) fixed-width confidence interval (FWCI) estimation and (ii) minimum risk point estimation (MRPE) problems for comparing the location parameters from two independent negative exponential (NE) populations with unknown location parameters but unknown and unequal scale parameters. The NE distributions have been used in the literature for studying the growth of certain kinds of tumors in cancer research as well as in survival and reliability analyses. We first develop FWCI estimation strategies for comparing the location parameters. We formulate a unified multi-stage estimation strategy and derive the theory for asymptotic second order (s.o.) expansions of the coverage probability as well as s.o. efficiency. Then, we successively specialize the general estimation strategy by providing a range of asymptotic analyses associated with (i) purely sequential, (ii) parallel piecewise sequential, (iii) accelerated sequential, and (iv) three-stage sampling. The work is supported using interesting simulated and real applications from cancer studies. Next, we develop a two-stage FWCI estimation strategy that customarily assures that the associated exact confidence coefficient is at least (1 − 𝛼), a nominal prescribed goal. We develop tables for the "exact" and the estimated values of the appropriate percentage points required to implement the two-stage estimation strategy, allowing both equal and unequal pilot sizes. These tables are invaluable for our problem and many other related problems, but these have largely remained nearly elusive. We wrap up with illustrations from simulated data and real data from cancer studies to highlight the theoretical findings of practical value. We then move on to develop minimum risk point estimation (MRPE) methodologies for comparing the location parameters. We start by designing a general theory of multi-stage sampling strategies under reasonable assumptions. This theory has led to important and practical properties such as (i) the asymptotic first-order (f.o.) risk efficiency and (ii) the asymptotic s.o. expansion of the regret. Then, we have successively delivered a wide range of asymptotics associated with specific multi-stage strategies such as (i) purely sequential, (ii) accelerated sequential, (iii) three-stage, and (iv) two-stage sampling methods. The work is supported with interesting illustrations using simulated and cancer data. All our proposed methodologies enjoy attractive asymptotic efficiency and consistency properties depending on each scenario under consideration. Finally, we conclude by adding a number of broad-ranging areas of future research, which we fully expect to be of practical significance and methodological advancements with considerable backing from theory.