The shape of Bayes tests of power one
Given a Brownian motion with unknown drift \(d\in {\mathbb{R}}\) and \(P_ d\) the associated measure, a test of power one for the decision problem \(H_ 0: d\neq 0\) versus \(H_ 1: d=0\) is by definition a stopping time T which satisfies the conditions \(P_ 0(t<\infty)<1\) and \(P_ d(T<\infty)=1\) if \(d\neq 0\). Here stopping also means a decision in favour of ''d\(\neq 0''\). If the unit sampling cost c depends on d in an appropriate way, it turns out that a simple Bayes rule is approximately optimal. Such a rule stops sampling when the posterior probability of the hypothesis is too small. The paper is organized as follows: Theorem 1 states the existence of an optimal (Bayes) stopping rule \(T_ c\). Theorem 2 gives upper and lower bounds for \(T_ c\) which make it possible to derive its asymptotic shape when c tends to zero. Theorem 3 refines these bounds, which yields an o(c)-approximation of the minimal Bayes risk (Theorem 4). Finally, Theorem 5 treats the one-sided case.
scientific article; zbMATH DE number 3976144