Sharp adaptive estimation of quadratic functionals

Abstract

Estimation of a quadratic functional of a function observed in the Gaussian white noise model is considered. A data-dependent method for choosing the amount of smoothing is given. The method is based on comparing certain quadratic estimators with each other. It is shown that the method is asymptotically sharp or nearly sharp adaptive simultaneously for the "regular" and "irregular" region. We consider l p bodies and construct bounds for the risk of the estimator which show that for p = 4 the estimator is exactly optimal and for example when p ∈ [3, 100], then the upper bound is at most 1.055 times larger than the lower bound. We show the connection of the estimator to the theory of optimal recovery. The estimator is a calibration of an estimator which is nearly minimax optimal among quadratic estimators.

Multiresolution local polynomial regression: A new approach to pointwise spatial adaptation

“…The convolutions (26) and (27) assume that the observations z(x) defined on the finite grid (2) are completed by zeros (zero padded) outside of this finite grid for the infinite regular grid.…”

“…where In what follows for the sake of simplicity we use the linear estimate given as the convolutions (26) and (27).…”

“…Some recent results concerning a development of the adaptive scale multivariable estimation can be seen in [26] where the optimal kernels are derived for different classes of functions to estimate.…”

Asymptotically exact minimax estimation in sup-norm for anisotropic Hölder classes

“…Such kernels of the form K(x) = (1 − x ) + , for x ∈ R d , are studied in Klemelä and Tsybakov (2001). We denote (t) + = max(0, t).…”

Deblocking Artifact of Satellite Image Based on Adaptive Soft-Threshold Anisotropic Filter Using Wavelet

“…A nonlinear estimator is applied, which is derived from a local polynomial approach of satellite image in a wavelet window. Generalized linear model [55], estimating regression [56], smoothing adaptation [57], sharp adaptive estimation [58], signal dependent noise (SDN) model [59], and an adaptive jump-preserving (AJP) estimation [60] discussed the kernel by using an adaptive estimation. Yet, they did not apply Lepski's methods as we did.…”