The spatially inhomogeneous smoothness of nonparametric methods is often modelled by Besov and Triebel-type smoothness constraints. For such problems, Donoho and Johnstone (1992) ([DoJo]), Delyon and Juditsky (1993) ([DJ]) studied minimax rates of convergence for wavelet estimators with thresholding, while Lepski, Mammen and Spokoiny (1995) ([LMS]) proposed a variable bandwidth selection for kernel estimators that achieved optimal rates over Besov classes. However, a second challenge in many applications of non-parametric curve estimation is that the function must also satisfy some (lower and/or upper) variable order constraints (for example, a density must be non-negative or a density is constrained to lie between two functions). In this work we show how to construct estimators under order constraints that satisfy these constraints and also achieve minimax rates over the appropriate smoothness class. In [DM1] we studied the case when the lower and/or upper constraints are constants; here we consider the general case when these constraints are functions. The parameters of the new constrained estimator are shown here to depend on the regularity of the constraint functions, except when the lower constraint function is convex and/or the upper constraint function is concave.
Paru en septembre 2001 , 27 pages