Let W1,W2,...be independent and identically distributed, positive and continuous, random variables having all moments finite with E(W1) = 0 Suppose thatwhere. Let N1(h) be the smallest integerand define N(h) = max{t(h), Nl(h)}. We sample the difference {N(h) - t(h)} in one single batch and term N(h) as the accelerated stopping time. First we derive asymptotic (as h → 0) second-order properties of various positive and negative moments of N(h) under appropriate conditions on m. Then, this machinery is used for various statistical problems in constructing fixed-size confidence regions and minimum risk point estimators. For some of these problems, we discuss moderate sample size performances of N(h) and associated estimators obtained through simulated experiments and contrast with those of the corresponding customary purely sequential and three - stage estimation procedures. With "proper" choices of of ρ and q, we observe that the accelerated version N(h) can save considerable amount of sampling operations and its moderate sample properties are (i) remarkably similar to those of the purely sequential procedures, and (ii) often better than those of the three-stage procedures.