WIAS Preprint No. 386, (1998)

Minimax detection of a signal for qn-balls with l_pn balls removed



Authors

  • Ingster, Yuri I.

2010 Mathematics Subject Classification

  • 62G10 62G20

Keywords

  • minimax hypotheses testing, asymptotics of error probabilities, infinitely divisible distributions

Abstract

In this paper we continue the researches of hypothesis testing problems leading to infinitely divisible distributions which have been started in the papers by Ingster, 1996a, 1997. Let the $n$-dimensional Gaussian random vector $x=xi+v$ is observed where $xi$ is a standard $n$-dimensional Gaussian vector and $vin R_n$ is an unknown mean. We consider the minimax hypothesis testing problem $H_0: v=0$ versus alternatives $H_1: vin V_n$, where $V_n$ is $l^n_q$-ball of radius $R_1,n$ with $l^n_p$-balls of radius $R_2,n$ removed. We are interesting in the asymptotics (as $n toinfty$) of the minimax second kind error probability $beta_n(alpha)=beta_n(alpha,p,q,R_ 1,n,R_2,n)$ where $alpha in (0,1)$ 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: $pi^n=pi_n times cdots times pi_n$. Here $pi_n= (1-h_n)delta_0+frach_n2(delta_-b_n+delta_b-n)$ are the three-point measures with some $h_n=h_n(p,q,R_1,n,R_2,n$ and $b_n=b_n(p,q,R_1,n,R_2,n$. 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 $ pleq q$, then the asymptotics of $beta_nalpha$ 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.

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