ISSN:
1432-2064
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Summary X 1,⋯,X〉 n are independent, identically distributed random variables with common density function f(x¦θ 1 ,⋯,θ k ,θ k+1 ), assumed to satisfy certain standard regularity conditions. The k+1 parameters are unknown, and the problem is to test the hypothesis that θ k+1 =b against the alternative that θ k+1 =b+cn −1/2 . θ 1 ,⋯,θ k are nuisance parameters. For this problem, the following artificial problem is temporarily substituted. It is known that ¦θ 1 -a i ¦≦n −1/2 M(n) for i=1,⋯,k, where a 1 , ⋯,a k are known, and M(n) approaches infinity as n increases but n −1/2 M(n) approaches zero as n increases. A Bayes decision rule is constructed for this artificial problem, relative to the a priori distribution which assigns weight A to θ k+1 =b, and weight 1-A to θ k+1 =b+cn −1/2 , in each case the weight being spread uniformly over the possible values of θ 1 ,⋯,θ k in the artificial problem. An analysis of the structure of the Bayes rule shows that if estimates of θ 1 ,...,θ k are substituted for a 1 ..., a k respectively, the resulting rule is a solution to the original problem, and this rule has the same asymptotic properties as a solution to the artificial problem as the Bayes rule for the artificial problem, no matter what the values a 1 ..., a k are.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00537521
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