WIAS Preprint No. 867, (2003)

Minimax nonparametric hypothesis testing for small type I errors



Authors

  • Ingster, Yuri I.
  • Suslina, Irina A.

2010 Mathematics Subject Classification

  • 62G10 62G20

Keywords

  • Minimax hypothesis testing, nonparametric signal detection, adaptive hypothesis testing, intermediate efficiency

Abstract

Under the white Gaussian noise model with the noise level $eto 0$, we study minimax nonparametric hypothesis testing problem $H_0 : f=0$ on unknown function $fin L_2(0,1)$. We consider alternative sets that are determined a regularity constraint in the Sobolev norm and we suppose that signals are bounded away from the null either in $L_2$-norm or in $L_infty$-norm. Analogous problems are considered in the sequence space. If type I error probability $ain (0,1)$ is fixed, then these problems were studied in book citeIS.02. In this paper we consider the case $ato 0$. We obtain either sharp distinguishability conditions or sharp asymptotics of the minimax type II error probability in the problem. We show that if $a$ is ``not too small'', then there exists natural extension of results citeIS.02, whenever if $a$ is ``very small'', then we obtain classical asymptotics and distinguishability conditions for small $a$. Adaptive problems are studied as well.

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