Adaptive minimax estimation of a fractional derivative
EURANDOM Technical Report
In this paper we consider a problem of adaption in estimating a fractional derivative of order $-1/2$ of an unknown density from observations in the Gaussian white noise. This problem is closely related to the Wicksell problem
of estimating an unknown distribution function of the radii of balls, based on their observed cross-sections.
We consider a similar problem of estimating an unknown infinitely dimensional vector $v(\theta)$ with components $\theta_k/\sqrt k$ from the observations of the signal $\theta$ in the white noise model. Under the assumption that $\theta$ belongs to a Sobolev ellipsoid with unknown smoothness, an adaptive minimax estimator of $v(\theta) $is constructed.