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December, 1991 Minimax Bayes Estimation in Nonparametric Regression
Nancy E. Heckman, Michael Woodroofe
Ann. Statist. 19(4): 2003-2014 (December, 1991). DOI: 10.1214/aos/1176348383

Abstract

One observes $n$ data points, $(\mathbf{t}_i, Y_i),$ with the mean of $Y_i$, conditional on the regression function $f,$ equal to $f(\mathbf{t}_i).$ The prior distribution of the vector $\mathbf{f} = (f(\mathbf{t}_1), \ldots, f(\mathbf{t}_n))^t$ is unknown, but lies in a known class $\Omega.$ An estimator, $\hat{\mathbf{f}},$ of $\mathbf{f}$ is found which minimizes the maximum $E\|\hat{\mathbf{f}} - \mathbf{f}\|^2.$ The maximum is taken over all priors in $\Omega$ and the minimum is taken over linear estimators of $\mathbf{f}.$ Asymptotic properties of the estimator are studied in the case that $\mathbf{t}_i$ is one-dimensional and $\Omega$ is the set of priors for which $f$ is smooth.

Citation

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Nancy E. Heckman. Michael Woodroofe. "Minimax Bayes Estimation in Nonparametric Regression." Ann. Statist. 19 (4) 2003 - 2014, December, 1991. https://doi.org/10.1214/aos/1176348383

Information

Published: December, 1991
First available in Project Euclid: 12 April 2007

zbMATH: 0747.62014
MathSciNet: MR1135161
Digital Object Identifier: 10.1214/aos/1176348383

Subjects:
Primary: 65D10

Keywords: Bayes estimates , Minimax estimates , Nonparametric regression , smoothing

Rights: Copyright © 1991 Institute of Mathematical Statistics

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Vol.19 • No. 4 • December, 1991
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