Adaptive Density Estimation on the Circle by Nearly Tight Frames

Abstract

This work is concerned with the study of asymptotic properties of nonparametric density estimates in the framework of circular data. The estimation procedure here applied is based on wavelet thresholding methods: the wavelets used are the so-called Mexican needlets, which describe a nearly-tight frame on the circle. We study the asymptotic behaviour of the L 2 -risk function for these estimates, in particular its adaptivity, proving that its rate of convergence is nearly optimal.

Quantitative central limit theorems for Mexican needlet coefficients on circular Poisson fields

“…Proposition 3.3. We will also propose a case study concerning the nonparametric density estimation, which can be considered as a completion of our previous work [18].…”

“…The interaction of these methods is successfully applied to exploit rates of convergence of the asymptotic normal approximation for functionals of Gaussian random in the spatial domain, stronger than the localization related to standard needlets (cfr. also [15,18]). Moreover, while the standard needlets are defined over a set of exact cubature points and weights (cfr.…”

“…[33]) to have a tight frame, the Theorem 2.2 in [22] establishes that the frame obtained by the Mexican needlets built over a set of points under some weaker conditions (see [22] and Section 2 below) is nearly-tight. Various examples of statistical applications of Mexican needlets can be found, for instance, in [42,28,32,18,14].…”

“…A complete overview on this topic can be found, for instance, in the textbooks [6,19,41,43]. Some more recent applications can be found in [1,10,18,27,47]. In recent years (cfr.…”

Adaptive global thresholding on the sphere

“…The asymptotic properties of other estimators for spherical data, not concerning the needlet framework, were investigated by Kim and Koo [31,32,33], while needlet-like nearly-tight frames were used in Durastanti [8] to establish the asymptotic properties of density function estimators on the circle. Finally, in Gautier and Le Pennec [15], the adaptive estimation by needlet thresholding was introduced in the nonparametric random coefficients binary choice model.…”