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Zbl 0815.62019
Donoho, David L.; Johnstone, Iain M.
Ideal spatial adaptation by wavelet shrinkage. (English)
[J] Biometrika 81, No.3, 425-455 (1994). ISSN 0006-3444; ISSN 1464-3510/e

Summary: With ideal spatial adaptation, an oracle furnishes information about how best to adapt a spatially variable estimator, whether piecewise constant, piecewise polynomial, variable knot spline, or variable bandwidth kernel, to the unknown function. Estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels; however, it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone. \par We describe a new principle for spatially-adaptive estimation: selective wavelet reconstruction. We show that variable-knot spline fits and piecewise-polynomial fits, when equipped with an oracle to select the knots, are not dramatically more powerful than selective wavelet reconstruction 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 in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately $2\log n$, where $n$ is the sample size. \par Moreover no estimator can give a better guarantee than this. Within the class of spatially adaptive procedures, RiskShrink is essentially optimal. Relying only on the data, it comes within a factor $\log\sp 2 n$ of the performance of piecewise polynomial and variable-knot spline methods equipped with an oracle. In contrast, it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone.

MSC 2000:: *62G07 Curve estimation
62C99 Statistical decision theory

Keywords: minimax estimation subject to doing well at a point; orthogonal wavelet bases of compact support; spatially-adaptive estimation; selective wavelet reconstruction; variable-knot spline fits; piecewise-polynomial fits; RiskShrink; shrinkage of empirical wavelet coefficients; oracle inequality

Cited in: Zbl 1170.42306 Zbl 1099.94516 Zbl 1085.65023 Zbl 1054.62035 Zbl 1043.94002 Zbl 0993.92023 Zbl 0980.62053 Zbl 1013.62045 Zbl 1069.62502 Zbl 0954.62040 Zbl 0953.62037 Zbl 0883.62038 Zbl 0845.62034 Zbl 0820.62002

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