## The Annals of Statistics

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

### Second Order Approximations for Sequential Point and Interval Estimation

#### 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