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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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