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 ε → 0, we study minimax nonparametric hypothesis testing problem 𝐻0 : ƒ = 0 on unknown function ƒ ∈ L2(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 L2-norm or in L-norm. Analogous problems are considered in the sequence space. If type I error probability α ∈ (0,1) is fixed, then these problems were studied in book [13]. In this paper we consider the case α → 0. We obtain either sharp distinguishability conditions or sharp asymptotics of the minimax type II error probability in the problem. We show that if α is "not too small", then there exists natural extension of results [13], whenever if α is "very small", then we obtain classical asymptotics and distinguishability conditions for small α. Adaptive problems are studied as well.

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