On estimation and detection of smooth high-dimensional function
- Ingster, Yuri I.
- Suslina, Irina
2010 Mathematics Subject Classification
- 62G10 62G20
- high-dimensional estimation, high-dimensional signal detection, minimax hypothesis testing, separation rates, Sobolev norms, lattice problem
Observing an unknown $n$-variables function $f(t), tin [0,1]^n$ in the white Gaussian noise of a level $e>0$. We suppose that there exist $1$-periodical (in each variable) $sigma$-smooth extensions of functions $f(t)$ on $R^n$ and $f$ belongs to a Sobolev ball, i.e., $ f _sigma,2leq 1$, where $ cdot _sigma,2$ is a Sobolev norm (we consider two variants of one). We study two problem: estimation of $f$ and testing of the null hypothesis $H_0: f=0$ against alternatives $ f _2geq r_e$. We study the asymptotics (as $eto 0, ntoinfty$) of the minimax risk for square losses, for estimation problem, and of minimax error probabilities and of minimax separation rates in the detection problem. We show that of $ntoinfty$, then there exist ``sharp separation rates'' in the detection problem. The asymptotics of minimax risks of estimation and of separation rates of testing are of different type for $nll loge^-1$ and for $ngg loge^-1$. The problems under consideration are closely related with ``lattice problem'' in the numerical theory.