Estimation and detection of functions from anisotropic Sobolev classes

By Yu. I. Ingster, N. Stepanova. 2011

28 citations recorded by the source corpus.

: We consider the problems of estimating and detecting an un- known 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.

Subfields: Applied Mathematics, Numerical Analysis, Statistics and Probability

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Numerical Analysis

In Statistics and Probability

Citations in the corpus

Fast Distribution To Real Regression (2013, neutral): “See [3, 6] for other analyses with this type of assumption.”
Estimating Cosmological Parameters from the Dark Matter Distribution (2016, background): “If the space of input densities, I , is in a Sobolov ellip-soid type space; see (Ingster & Stepanova, 2011; Laurent, 1996; Oliva et al., 2014) for details.”
Fast Function to Function Regression (2014, unknown): “D = {(Pi, Qi)}i=1 where (3) Pi = {pi(uij) + ij} j=1 , Qi = {qi(vij) + ξij} mi j=1 , (4)”
Distribution to Distribution Regression (2013, background): “With (7), (2) we can upperbound the L2 loss of f̂(p̃0): ‖f̂(p̃0)− f(p0)‖2 = = (∫ Λk ( ∑ α∈At (âα − aα(f(P0)))ϕα(x) − ∑ α∈Act aα(f(P0))ϕα(x) )2 dx )1/2 = √∑ α∈At (âα − aα(f(P0)))2 + ∑ α∈Act a2α(f(P0)) (8) ≤ ∑ α∈At |âα − aα(f(P0))|+ ∑ α∈Act a2α(f(P0)) 1/2 where (8) follows from orthonormality.”
Tight conditions for consistency of variable selection in the context of high dimensionality (2011, background): “The first term in this maximum coincides, up to the logarithmic term, with the minimax rate of separation in the problem of detection of an s -dimensional signal [22].”