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Statistica Sinica 20 (2010), 1011-1024





SHARP MINIMAX ESTIMATION OF THE VARIANCE

OF BROWNIAN MOTION CORRUPTED

WITH GAUSSIAN NOISE


T. Tony Cai$^1$, A. Munk$^2$ and J. Schmidt-Hieber$^2$


$^1$University of Pennsylvania and $^2$Universität Göttingen


Abstract: Let $W_t$ be a Brownian motion with $\epsilon_{in}\stackrel{i.i.d.}{\sim}\mathcal{N}(0,1)$, $i=1,\ldots,n,$ independent of $W_t$. $\sigma, \tau>0$ are real, unknown parameters. Suppose we observe $Y_{i,n}=\sigma W_{i/n}+\tau
\epsilon_{in}.$ In this paper we establish sharp estimators for $\sigma^2$ and $\tau^2$ in minimax sense, i.e. they attain the minimax constant asymptotically. A short and direct proof for the minimax lower bound is given. These estimators are based on a spectral decomposition of the underlying process $Y_{i,n}$ and can be computed explicitly taking $O(n\log n)$ operations. We outline how these estimators can be generalized from Brownian motion to processes with independent increments. Further we show that the spectral estimators presented are asymptotically normal.



Key words and phrases: Asymptotic normality, Brownian motion, deconvolution, minimax, spectral estimators, statistical inverse problems, variance estimation, oracle estimator.

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