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 $eto 0$, 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.