Renormalizing upper and lower bounds for integrated risk in the white noise model
The author considers the problem of estimation of the function \(f\) in the white noise model \[ dX_ t= f(t)dt+\sigma dW_ t, \quad 0\leq t\leq 1, \qquad f\in\mathbb{F}\subset \mathbb{L}^ 2[0,1]. \] He develops an original method to obtain upper and lower bounds for the minimax risk \[ \mathbb{R} (\mathbb{F},\sigma)= \inf_ \delta \sup_{f\in\mathbb{F}} \mathbb{E} \int_ 0^ 1 (f(x)- \delta(x))^ 2 dx. \] This is done for non-ellipsoidal sets \(\mathbb{F}\), extending results of Pinsker and Efroimovich, and Pinsker. In his proof he uses the invariance properties of \(\mathbb{F}\). As an example he considers \(\mathbb{F}\) as the set of functions having \((k-2)\) continuous derivatives and for which the \((k-1)\) derivative is Lipschitz with constant \(M\) and periodically continuous. He obtains in this case: \[ 0,196 M^{2/3}\leq \limsup_{\sigma\to 0} \sigma^{-2/3} \mathbb{R}(\mathbb{F}, \sigma)\leq M^{2/3}/ 3^{1/3}. \]
scientific article; zbMATH DE number 444768