AbstractConsider a given sequence {Tn} of estimators for a real‐valued parameter θ. This paper studies asymptotic properties of restricted Bayes tests of the following form: rejectH0:θ ≤ θ0in favour of the alternative θ>θ0ifTn≤Cn, where the critical pointCnis determined to minimize among all tests of this form the expected probability of error with respect to the prior distribution. Such tests may or may not be fully Bayes tests, and so are calledTn‐Bayes. Under fairly broad conditions it is shown that\documentclass{article}\pagestyle{empty}\begin{document}$$ c_n = \theta _0 + a_n b(\theta _0)\bar \mu + 0(a_n) $$\end{document}and theTn‐Bayes risk\documentclass{article}\pagestyle{empty}\begin{document}$$ B_n (c_n) = a_n b(\theta _0)p(\theta _0)p(\theta _0)\int_{ - \infty }^\infty {|x - \bar \mu |dF(x) + 0(a_n)} $$\end{document}whereanis the order of the standard error ofTn, ‐ is the prior density, andμis the median ofF, the limit distribution of (Tn– θ)/anb(θ). Sever