Summary
We study the problem of estimating an unknown function on the unit interval (or itsk-th derivative), with supremum norm loss, when the function is observed in Gaussian white noise and the unknown function is known only to obey Lipschitz-β smoothness, β>k≧0. We discuss an optimization problem associated with the theory ofoptimal recovery. Although optimal recovery is concerned with deterministic noise chosen by a clever opponent, the solution of this problem furnishes the kernel of the minimax linear estimate for Gaussian white noise. Moreover, this minimax linear estimator is asymptotically minimax among all estimates. We sketch also applications to higher dimensions and to indirect measurement (e.g. deconvolution) problems.
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Dedicated to R.Z. Khas'minskii for his 60th birthday
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Donoho, D.L. Asymptotic minimax risk for sup-norm loss: Solution via optimal recovery. Probab. Th. Rel. Fields 99, 145–170 (1994). https://doi.org/10.1007/BF01199020
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DOI: https://doi.org/10.1007/BF01199020