Neo-Classical Minimax Problems , Thresholding , and Adaptation

We study the problem of estimating from data Y N( ; ) under squared-error loss. We de ne three new scalar minimax problems in which the risk is weighted by the size of . Simple thresholding gives asymptotically minimax estimates of all three problems. We indicate the relationships of the new problems to each other and to two other neo-classical problems: the problems of the bounded normal mean and of the risk-constrained normal mean. Via the wavelet transform, these results have implications for adaptive function estimation, to: (1) estimating functions of unknown type and degree of smoothness in a global ` norm; (2) estimating a function of unknown degree of local Holder smoothness at a xed point. In setting (2), the scalar minimax results imply: (a) that it is not possible to fully adapt to unknown degree of smoothness { adaptation imposes a performance cost; and (b) that simple thresholding of the empirical wavelet transform gives an estimate of a function at a xed point which is, to within constants, optimally adaptive to unknown degree of smoothness.