The Annals of Statistics

Second Order Approximations for Sequential Point and Interval Estimation

Michael Woodroofe

Full-text: Open access

Abstract

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.

Article information

Source
Ann. Statist., Volume 5, Number 5 (1977), 984-995.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176343953

Digital Object Identifier
doi:10.1214/aos/1176343953

Mathematical Reviews number (MathSciNet)
MR494735

Zentralblatt MATH identifier
0374.62081

JSTOR
links.jstor.org

Subjects
Primary: 62L12: Sequential estimation
Secondary: 60F05: Central limit and other weak theorems

Keywords
Excess under the boundary sequential estimation fixed width confidence intervals

Citation

Woodroofe, Michael. Second Order Approximations for Sequential Point and Interval Estimation. Ann. Statist. 5 (1977), no. 5, 984--995. doi:10.1214/aos/1176343953. https://projecteuclid.org/euclid.aos/1176343953


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