Canadian Journal of Statistics

Asymptotic properties of Bayes risk for one‐sided tests

Journal Article


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

  • equation image

and the Tn‐Bayes risk

  • equation image

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.

Related Topics

Related Publications

Related Content

Site Footer


This website is provided by John Wiley & Sons Limited, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ (Company No: 00641132, VAT No: 376766987)

Published features on are checked for statistical accuracy by a panel from the European Network for Business and Industrial Statistics (ENBIS)   to whom Wiley and express their gratitude. This panel are: Ron Kenett, David Steinberg, Shirley Coleman, Irena Ograjenšek, Fabrizio Ruggeri, Rainer Göb, Philippe Castagliola, Xavier Tort-Martorell, Bart De Ketelaere, Antonio Pievatolo, Martina Vandebroek, Lance Mitchell, Gilbert Saporta, Helmut Waldl and Stelios Psarakis.