## Renormalizing upper and lower bounds for integrated risk in the white noise model.(English)Zbl 0795.62038

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: $0,196 M^{2/3}\leq \limsup_{\sigma\to 0} \sigma^{-2/3} \mathbb{R}(\mathbb{F}, \sigma)\leq M^{2/3}/ 3^{1/3}.$

### MSC:

 62G07 Density estimation 62G20 Asymptotic properties of nonparametric inference 62M05 Markov processes: estimation; hidden Markov models
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