Second-order asymptotics in a class of purely sequential minimum risk point estimation (MRPE) methodologies

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Under the squared error loss plus linear cost of sampling, we revisit the minimum risk point estimation (MRPE) problem for an unknown normal mean $$\mu$$ when the variance $$\sigma ^{2}$$ also remains unknown. We begin by defining a new class of purely sequential MRPE methodologies based on a general estimator $$W_{n}$$ for $$\sigma$$ satisfying a set of conditions in proposing the requisite stopping boundary. Under such appropriate set of sufficient conditions on $$W_{n}$$ and a properly constructed associated stopping variable, we show that (i) the normalized stopping time converges in law to a normal distribution (Theorem 3.3), and (ii) the square of such a normalized stopping time is uniformly integrable (Theorem 3.4). These results subsequently lead to an asymptotic second-order expansion of the associated regret function in general (Theorem 4.1). After such general considerations, we include a number of substantial illustrations where we respectively substitute appropriate multiples of Gini’s mean difference and the mean absolute deviation in the place of the general estimator $$W_{n}$$ . These illustrations show a number of desirable asymptotic first-order and second-order properties under the resulting purely sequential MRPE strategies. We end this discourse by highlighting selected summaries obtained via simulations.