Footnote:
Diese Datenquelle enthält auch Bestandsnachweise, die nicht zu einem Volltext führen.
Description:
In this paper we continue the researches of hypothesis testing problems leading to innitely divisible distributions which have been started in the papers by Ingster, 1996a, 1997. Let the n-dimensional Gaussian random vector x = ξ + v is observed where ξ is a standard n-dimensional Gaussian vector and v ∈ 𝑅n is an unknown mean. We consider the minimax hypothesis testing problem H0 : v = 0 versus alternatives H1 : v ∈ Vn, where Vn is lnq-ball of radius R1,n with lnp-balls of radius R2,n removed. We are interesting in the asymptotics (as n → ∞) of the minimax second kind error probability βn(α) = βn(α; p, q, R1,n, R2,n) where α ∈ (0,1) is a level of the first kind error probability. Close minimax estimation problem had been studied by Donoho and Johnstone (1994). We show that the asymptotically least favorably priors in the problem of interest are of the product type: πn = πn × … × n. Here πn = (1 − hn)δ0 + hn ⁄2 (δ − bn + δ b − n) are the three-point measures with some hn = hn(p, q, R1,n, R2,n and bn = bn(p, q, R1,n, R 2,n. This reduces the problem of interest to Bayssian hypothesis testing problems where the asymptotics of error probabilities had been studied by Ingster, 1996a, 1997. In particularly, if p ≤ q, then the asymptotics of n are of Gaussian type, but if p > q then its are either Gaussian or degenerate or belong to a special class of infinitely divisible distributions which was described in Ingster, 1996a, 1997.