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Bounds for the Bayes risk for testing sequentially the sign of the drift parameter of a Wiener process. (English) Zbl 0543.62057

Let x(t) be a Wiener process with drift \(\mu\) and variance 1 per unit time. Consider the following problem: 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 \(c=\cos t\) of observation per unit time, and \(\mu\) has a prior distribution which is normal with mean 0 and variance \(\sigma^ 2_ 0.\)
The authors obtain a lower bound for the Bayes risk for the case of \(\sigma_ 0\) finite, and show that this lower bound is not asymptotically achievable as \(\sigma_ 0\to \infty\) for all \(c>0\). Furthermore, they show that the P. J. Bickel and J. A. Yahav’s lower bound [Tech. Rep. No.26, Dept. Statist., Stanford Univ. (1967)] is not asymptotically achievable as \(c\to 0\) for the case of \(\mu\) having the improper prior distribution.
Reviewer: K.Uosaki

MSC:

62L10 Sequential statistical analysis
62C10 Bayesian problems; characterization of Bayes procedures
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