Second Order Approximations for Sequential Point and Interval Estimation

Several stopping times which arise from problems of sequential estimation may be written in the form $t_c = \inf\{n \geqq m: S_n < cn^\alpha L(n)\}$ where $S_n, n \geqq 1,$ are the partial sums of i.i.d. positive random variables, $\alpha > 1, L(n)$ is a convergent sequence, and $c$ is a positive parameter which is often allowed to approach zero. In this paper we find the asymptotic distribution of the excess $R_c = ct_c^\alpha - S_{t_c}$ as $c \rightarrow 0$ and use it to obtain sharp estimates for $E\{t_c\}.$ We then apply our results to obtain second order approximations to the expected sample size and risk of some sequential procedures for estimation.Several stopping times which arise from problems of sequential estimation may be written in the form $t_c = \inf\{n \geqq m: S_n < cn^\alpha L(n)\}$ where $S_n, n \geqq 1,$ are the partial sums of i.i.d. positive random variables, $\alpha > 1, L(n)$ is a convergent sequence, and $c$ is a positive parameter which is often allowed to approach zero. In this paper we find the asymptotic distribution of the excess $R_c = ct_c^\alpha - S_{t_c}$ as $c \rightarrow 0$ and use it to obtain sharp estimates for $E\{t_c\}.$ We then apply our results to obtain second order approximations to the expected sample size and risk of some sequential procedures for estimation.

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Second Order Approximations for Sequential Point and Interval Estimation

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