On general asymptotically second-order efficient purely sequential fixed-width confidence interval (FWCI) and minimum risk point estimation (MRPE) strategies for a normal mean and optimality

Abstract

We develop a generalized class of purely sequential sampling strategies associated with both fixed-width confidence interval (FWCI) and minimum risk point estimation (MRPE) problems for the unknown mean $$\mu $$ of a normally distributed population having its variance $$\sigma ^{2}$$ also unknown. Under this newly proposed general class of associated estimation strategies, we develop a variety of asymptotic first-order and asymptotic second-order properties such as asymptotic consistency, first-order efficiency, first-order risk efficiency, second-order efficiency, and second-order regret analysis. Next, we proceed to locate an optimal strategy within our newly built large class of possibilities. Such optimality is defined as having been associated with the minimal second-order asymptotic variance of a stopping time within the general class of proposed strategies. We follow through by exploring both the FWCI and MRPE problems with the help of data analysis from simulations.