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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 when the variance 2 also remains unknown. We begin by defining a new class of purely sequential MRPE methodologies based on a general estimator W n for 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 secondorder properties under the resulting purely sequential MRPE strategies. We end this discourse by highlighting selected summaries obtained via simulations.

The gamma distribution is a flexible right-skewed distribution widely used in many areas, and it is of great interest to estimate the probability of a random variable exceeding a specified value in survival and reliability analysis. Therefore, the study develops a fixed-accuracy confidence interval for P(X > c) when X follows a gamma distribution, Γ(α, β), and c is a preassigned positive constant through: 1) a purely sequential procedure with known shape parameter α and unknown rate parameter β; and 2) a nonparametric purely sequential procedure with both shape and rate parameters unknown. Both procedures enjoy appealing asymptotic first-order efficiency and asymptotic consistency properties. Extensive simulations validate the theoretical findings. Three real-life data examples from health studies and steel manufacturing study are discussed to illustrate the practical applicability of both procedures.

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