## Asymptotic properties of Bayes risk for one‐sided tests

### Abstract

Consider 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: reject H0:θ ≤ θ0 in favour of the alternative θ > θ0 if TnCn, where the critical point Cn is 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 called Tn‐Bayes. Under fairly broad conditions it is shown that

and the Tn‐Bayes risk

where an is the order of the standard error of Tn, ‐ is the prior density, and μ is the median of F, the limit distribution of (Tn – θ)/anb(θ). Several examples are given.

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