Minimax nonparametric hypothesis testing for small type I errors
- Ingster, Yuri I.
- Suslina, Irina A.
2010 Mathematics Subject Classification
- 62G10 62G20
- Minimax hypothesis testing, nonparametric signal detection, adaptive hypothesis testing, intermediate efficiency
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 . 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 , whenever if α is "very small", then we obtain classical asymptotics and distinguishability conditions for small α. Adaptive problems are studied as well.