Groupe d’études et de recherche en analyse des décisions


Asymptotically Minimax Non-Parametric Function Estimation with Positivity Constraints I


The spatially inhomogeneous smoothness of the non-parametric density or regression-function to be estimated by non-parametric methods is often modelled by Besov- and Triebel-type smoothness constraints. For such problems, Donoho and Johnstone (1992), Delyon and Juditsky (1993) studied minimax rates of convergence for wavelet estimators with thresholding, while Lepskii, Mammen and Spokoiny (1995) proposed a variable bandwidth selection for kernel estimators that achieved optimal rates over Besov classes. However, a second challenge in many real applications of non-parametric curve estimation is that the function must be positive. Here, we show how to construct estimators under positivity constraints that satisfy these constraints and also achieve minimax rates over the appropriate smoothness class.

, 27 pages