On some problems of hypothesis testing leading to infinitely divisible distributions
- Ingster, Yuri I.
2010 Mathematics Subject Classification
- 62G10 62G20
- Bayesian hypotheses testing, minimax hypotheses testing, asymptotics of error probabilities, infinitely divisible distributions
We observe an n-dimensional Gaussian random vector x ＝ ξ + v where ξ is a standard n-dimensional Gaussian vector and v ∈ Rn is an unknown mean and we consider the hypothesis testing problem H0 : v ＝ 0 against two related types of alternatives: Bayesian: the coordinates of v may be equal to -b, 0 or +b only and the number of nonzero coordinates is random with binomial distribution Bi(hn,n); Minimax: the coordinates of v may be equal to -b, 0 or +b only and the number, k, of nonzero coordinates is nonrandom. The values b ＝ bn > 0, h ＝ hn ∈ (0, 1] or an integer k ＝ kn ∈ [1,n] are given. These problems are of importance for many applications, for example for multi-channel detection and communication systems. We study the asymptotics of the log-likelihood distribution for Bayesian alternatives and show that they are either Gaussian or degenerate or belong to a special two-parametric class of infinitely divisible distributions. The latter corresponds to the case bn ≍ √log n and hn is small enough. We also show that randomization in the Bayesian alternative corresponds to asymptotically least favorable priors for minimax alrernative if nhn ＝ kn → ∞.