Sharp Adaptive Nonparametric Hypothesis Testing for Sobolev Ellipsoids

• Activity Number: 576
• Type: Contributed
• Date/Time: Wednesday, August 1, 2012 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Pengsheng Ji*+ and Michael Nussbaum
• Companies: and Cornell University
• Keywords: minimax hypothesis testing ; nonparametric signal detection ; sharp asymptotic adaptivity ; moderate deviation

Abstract

We consider testing for presence of a signal in Gaussian white noise with intensity $n^{-1/2}$, when the alternatives are given by smoothness ellipsoids with an $L_{2}$-ball of radius $\rho$ removed. It is known that, for a fixed Sobolev type ellipsoid $\Sigma(\beta,P)$ of smoothness $\beta$ and size $P$, the radius rate $\rho\asymp n^{-4\beta/(4\beta+1)}$ is the critical separation rate, and the sharp asymptotics of the minimax error of second kind at the separation rate is also found. For adaptation over both $\beta$ and $P$ in that context, it is known that a $\log\log$-penalty over the separation rate for $\rho$ is necessary for a nonzero asymptotic power. Here, following an example in nonparametric estimation related to the Pinsker constant, we investigate the adaptation problem over the ellipsoid size $P$ only, for fixed smoothness degree $\beta$. It is established that the Ermakov type sharp asymptotics can be preserved in that adaptive setting, if $\rho\rightarrow0$ slower than the separation rate. The penalty for adaptation in that setting turns out to be a sequence tending to infinity arbitrarily slowly.