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Minimax Bayes estimation in nonparametric regression. (English) Zbl 0747.62014

Summary: One observes \(n\) data points, \((t_ i,Y_ i)\), with the mean of \(Y_ i\), conditional on the regression function \(f\), equal to \(f(t_ i)\). The prior distribution of the vector \(f=(f(t_ 1),\dots,f(t_ n))'\) is unknown, but lies in a known class \(\Omega\). An estimator, \(\hat f\), of \(f\) is found which minimizes the maximum \(E\|\hat f-f\|^ 2\). The maximum is taken over all priors in \(\Omega\) and the minimum is taken over linear estimators of \(f\). Asymptotic properties of the estimator are studied in the case that \(t_ i\) is one-dimensional and \(\Omega\) is the set of priors for which \(f\) is smooth.

MSC:

62C20 Minimax procedures in statistical decision theory
62C10 Bayesian problems; characterization of Bayes procedures
62G07 Density estimation
65D10 Numerical smoothing, curve fitting
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