Sankhya: The Indian Journal of Statistics

2001, Volume 63, Series B, Pt. 2, pp. 149--180

NUMERICAL METHODS FOR ASYMPTOTICALLY MINIMAX NON-PARAMETRIC FUNCTION ESTIMATION WITH POSITIVITY CONSTRAINTS I

By

LUBOMIR DECHEVSKY, Université de Montréal, Canada
BRENDA MACGIBBON, Université du Québec á Montréal, Canada

and

SPIRIDON PENEV, University of New South Wales, Australia

SUMMARY. One important challenge in nonparametric density and regression-function estimation is spatially inhomogeneous smoothness. This is often modelled by Besov-type smoothness constraints. With this type of constraint, Donoho and Johnstone (1992), Delyon and Juditsky (1993) studied asymptotic-minimax optimal wavelet estimators with thresholding, while Lepski, Mammen and Spokoiny (1995) proposed a variable-bandwidth selection for kernel estimators that also achieved the asymptotic-minimax rates. However, a second challenge in many applications of nonparametric curve estimation is that the function must be nonnegative or order-constrained. Dechevsky and MacGibbon (1999) constructed wavelet- and kernel-based estimators under positivity constraints that satisfied these constraints and also achieved asymptotic-minimax rates over the appropriate smoothness classes. Here we show how to replace the integral in their definition by a quadrature formula in order to numerically construct the estimators, so that the new ``quadrature'' estimators enjoy the positivity- and smoothness-preserving properties of the ones in \DechM, and are also asymptotic-minimax optimal

AMS (2000) subject classification. Primary 62G07, 62G08, 65C60; secondary 06A05, 06A06, 41A29, 46B42, 65D30.

Key words and phrases. Density estimation, nonparametric regression, wavelet estimator, kernel estimator, order constraint, asymptotic minimax rate, numerical integration.

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