IDEAL SPATIAL ADAPTATION BY WAVELET SHRINKAGE

With ideal spatial adaptation, an oracle furnishes information about h ow best to adapt a spatially variable estimator, whether piecewise con stant, piecewise polynomial, variable knot spline, or variable bandwid th kernel, to the unknown function. Estimation with the aid of an orac le offers dramatic advantages over traditional linear estimation by no nadaptive kernels; however, it is a priori unclear whether such perfor mance can be obtained by a procedure relying on the data alone, We des cribe a new principle for spatially-adaptive estimation: selective wav elet reconstruction. We show that variable-knot spline fits and piecew ise-polynomial fits, when equipped with an oracle to select the knots, are not dramatically more powerful than selective wavelet reconstruct ion with an oracle. We develop a practical spatially adaptive method, RiskShrink, which works by shrinkage of empirical wavelet coefficients . RiskShrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so. A new inequality i n multivariate normal decision theory which we call the oracle inequal ity shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n, where n is the sample size. Moreover no estimator can give a better guarantee than this. Within t he class of spatially adaptive procedures, RiskShrink is essentially o ptimal. Relying only on the data, it comes within a factor log(2) n of the performance of piecewise polynomial and variable-knot spline meth ods equipped with an oracle. In contrast, it is unknown how or if piec ewise polynomial methods could be made to function this well when deni ed access to an oracle and forced to rely on data alone.