Groupe d’études et de recherche en analyse des décisions

G-2000-33

Numerical Methods for Asymptotically Minimax Non-parametric Function Estimation with Positivity Constraints I

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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. In [DM1] 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 [DM1], and are also asymptotic-minimax optimal.

, 47 pages