Second-order asymptotics for comparing treatment means from purely sequential estimation strategies under possible outlying observations

Abstract

We consider two independent populations, and with all four parameters assumed unknown, and In this paper, our goal is to revisit sequential minimum risk point estimation (MRPE) problems for the parameter say) under the squared error loss (SEL) plus linear cost of sampling in our quest for comparing two treatments. We explicitly consider two separate scenarios: (i) but is assumed unknown, and (ii) are unknown and unequal. In defining the relevant purely sequential stopping rules, customarily one incorporates multiples of sample standard deviations to set up requisite boundary crossing conditions. We set out to replace the multiples of sample standard deviations used in defining requisite boundary crossing conditions with Gini’s mean difference (GMD), mean absolute deviation (MAD), along with a number of combinations of the sample standard deviations, the GMD’s, and the MAD’s. We prove that the ensuing body of such newly developed purely sequential MRPE strategies have associated second-order regret expansions and that the GMD-based as well as the MAD-based strategies are better equipped to withstand occurrences of possible outlying observations. Our theory and methodologies are amply supported by both large-scale simulations and the illustrations with real data.