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{1176346729,
author = {Mallik, Ashim and Yao, Yi-Ching},
title = {Bounds for the Bayes Risk for Testing Sequentially the Sign of the Drift Parameter of a Wiener Process},
journal = {Ann. Statist.},
volume = {12},
number = {1},
year = {1984},
pages = { 1117-1123},
language = {en},
url = {http://dml.mathdoc.fr/item/1176346729}
}
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., Tome 12 (1984) no. 1, pp. 1117-1123. http://gdmltest.u-ga.fr/item/1176346729/