## The Annals of Statistics

- Ann. Statist.
- Volume 12, Number 3 (1984), 1117-1123.

### Bounds for the Bayes Risk for Testing Sequentially the Sign of the Drift Parameter of a Wiener Process

#### Abstract

Let $x(t)$ be a Wiener process with drift $\mu$ and variance 1 per unit time. The following problem is treated; test $H:\mu \leq 0$ vs. $A:\mu > 0$ with the loss function $|\mu|$ if the wrong decision is made and 0 otherwise, and with $c =$ cost of observation per unit time, where $\mu$ has a prior distribution which is normal with mean 0 and variance $\sigma^2_0$. An idea of Bickel and Yahav is followed to obtain a lower bound for the Bayes risk which is strict as $\sigma_0 \rightarrow \infty$ for all $c$. An upper bound is also derived.

#### Article information

**Source**

Ann. Statist., Volume 12, Number 3 (1984), 1117-1123.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176346729

**Digital Object Identifier**

doi:10.1214/aos/1176346729

**Mathematical Reviews number (MathSciNet)**

MR751300

**Zentralblatt MATH identifier**

0543.62057

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62L10: Sequential analysis

Secondary: 62C10: Bayesian problems; characterization of Bayes procedures

**Keywords**

Sequential tests S.P.R.T. Bayes stopping times lower bound asymptotic expansion

#### Citation

Mallik, Ashim; Yao, Yi-Ching. Bounds for the Bayes Risk for Testing Sequentially the Sign of the Drift Parameter of a Wiener Process. Ann. Statist. 12 (1984), no. 3, 1117--1123. doi:10.1214/aos/1176346729. https://projecteuclid.org/euclid.aos/1176346729