We consider the problems of estimating and detecting an unknown function f depending on a multidimensional variable (for instance, an image) observed in the Gaussian white noise. It is assumed that f be-longs to anisotropic Sobolev class. The case of a function of infinitely many variables is also considered. An asymptotic study (as the noise level tends to zero) of the estimation and detection problems is done. In connection with the estimation problem, we construct asymptotically minimax estima-tors and establish sharp asymptotics for the minimax integrated squared risk. In the detection problem, we construct asymptotically minimax tests and provide conditions for distinguishability in the problem.

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Keywords Anisotropic smoothness, Gaussian white noise, Multivariate functions, Nonparametric estimation, Nonparametric sig-nal detection
Persistent URL dx.doi.org/10.1214/11-EJS615
Journal Electronic Journal of Statistics
Ingster, Y. (Yuri), & Stepanova, N. (2011). Estimation and detection of functions from anisotropic Sobolev classes. Electronic Journal of Statistics, 5, 484–506. doi:10.1214/11-EJS615