Purely sequential FWCI and MRPE problems for the mean of a normal population by sampling in groups with illustrations using breast cancer data

Abstract

Abstract 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 (μ) 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, instead of gathering one observation at a time, we consider sequentially sampling k observations at a time in defining our proposed estimation strategies. We replace the customary sample standard deviation as an estimator for σ with a number of other pertinent estimators to come up with new and more appropriate stopping rules to suit the occasion. We do so because in real life we know that packaged items purchased in bulk often cost less per unit sample than the cost of an individual item. This article builds the whole array of estimation methodologies in order to address both FWCI and MRPE problems with appropriate first-order and second-order asymptotic analyses. These are followed by extensive sets of carefully laid out data analyses assisted via large-scale computer simulations. These are wrapped up with illustrations using breast cancer data.