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Multivariate resistant regression splines for estimating multivariate functions from noisy data. (English) Zbl 0906.62036
Summary: The multivariate resistant regression spline (MURRS) method for estimating an underlying smooth \(J\)-variate function by using noisy data is based on approximating it with tensor products of B-splines and minimizing a sum of the \(\rho\)-functions of the residuals to obtain a robust estimator of the regression function, where the spline knots are automatically chosen through a parallel of information criterion. When the knots are deterministically given, it is proved that the MURRS estimator achieves the optimal global convergence rates established by C. J. Stone [Ann. Statist. 13, 689-705 (1985; Zbl 0605.62065)] under some mild conditions. Examples are given to illustrate the utility of the proposed methodology. Usually, only a few tensor products of B-splines are enough to fit even complicated functions.
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62H12 Estimation in multivariate analysis
65D07 Numerical computation using splines
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