## The Annals of Statistics

- Ann. Statist.
- Volume 4, Number 4 (1976), 685-711.

### On Asymptotically Optimal Sequential Bayes Interval Estimation Procedures

Leon Jay Gleser and Sudhakar Kunte

#### Abstract

A theory of sequential Bayes interval estimation procedures for a single parameter is developed for the case where the loss for using an interval $I$ is a linear combination of the length of $I$, the indicator of noncoverage of $I$, and the number of observations taken. A class of stopping rules $\{t(c): c > 0\}$ is shown to be asymptotically pointwise optimal (A.P.O.) and asymptotically optimal (A.O.) for the confidence interval problem as the cost $c$ per observation tends to 0. The results require generalization of Bickel and Yahav's (1968) general conditions for the existence of A.P.O. and A.O. stopping rules to the case where the terminal risk $Y_n$ satisfies $f(n)Y_n \rightarrow V$ for $f(n)$ a regularly varying function on the integers.

#### Article information

**Source**

Ann. Statist., Volume 4, Number 4 (1976), 685-711.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176343542

**Digital Object Identifier**

doi:10.1214/aos/1176343542

**Mathematical Reviews number (MathSciNet)**

MR426323

**Zentralblatt MATH identifier**

0339.62059

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62L12: Sequential estimation

Secondary: 62L15: Optimal stopping [See also 60G40, 91A60] 62F15: Bayesian inference 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 62F10: Point estimation 62F25: Tolerance and confidence regions 60F99: None of the above, but in this section

**Keywords**

Bayes interval estimation procedures sequential inference stopping rules asymptotically pointwise optimal asymptotically optimal moderate deviation theory regularly varying function

#### Citation

Gleser, Leon Jay; Kunte, Sudhakar. On Asymptotically Optimal Sequential Bayes Interval Estimation Procedures. Ann. Statist. 4 (1976), no. 4, 685--711. doi:10.1214/aos/1176343542. https://projecteuclid.org/euclid.aos/1176343542