Asymptotically Minimax Non { Parametric Functionestimation with Positivity Constraints

D epartement de math ematiques Universit e du Qu ebec a Montr eal 0 PRELIMINARIES 1 Abstract One important challenge in non{parametric density and regression{function estimation is spatially inhomogeneous smoothness. This is often modelled by Besov-and Triebel{type smoothness constraints. 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. In a sequence of papers, of which this is the rst one, we show how to construct estimators under positivity constraints that satisfy these constraints and also achieve minimax rates over the appropriate smoothness class. 0 Preliminaries In this section we provide a list of some basic notation, deenitions and analytic results used later in the text. ] { integer part of (for > 0). x + = maxf0; xg. (0) = fw : 0; 0 ] ! 0; 1); !(t 1) > !(t 2); t 1 > t 2 ; 9 lim t!0 + !(t) = !(0 +) = 0; !(0) = 0g: Quasi-norm (see Bergh et al. (1976)], Dechevsky et al. (1997)]) : 9c = 1; ka+bk 5 c(kak+kbk). Semi-norm : the norm property kak = 0 , a = 0 is not necessarily fulllled. A { quasi(semi)normed abelian group { see Bergh et al. (1976)]. A { quasi-Banach space, if A is a complete quasi{normed abelian group, a linear space, and the quasinorm is homogeneous: kak A = jjkak A. A , ! B : continuous embedding of A in B (for quasinormed abelian groups) { see Bergh et al. (1976)]. A+B : A, B { quasinormed abelian groups: kak A+B = inf a=+ fkk A +kk B g (see Bergh et al. (1976)]). 0 PRELIMINARIES 2 K-functional: K(t; a; A; B) = inf a=+ ? kk A +tkk B , 0 < t < 1, (equivalent quasinorm in A+B { see Bergh et al. (1976)], Johnen et al. (1977)] for details and properties). B s pq (R d) { the inhomogeneous and homogeneous Besov spaces, 0 < p 5 1, 0 < q 5 1, s 2 R { see Bergh et al. For the properties of B s pq and F s pq we refer to Bergh et …