$\mathbb{L}_{p}$ adaptive estimation of an anisotropic density under independence hypothesis

Abstract

International audienceIn this paper, we focus on the problem of a multivariate density estimation under an Lp-loss. We provide a data-driven selection rule from a family of kernel estimators and derive for it Lp-risk oracle inequalities depending on the value of p ≥ 1. The proposed estimator permits us to take into account approximation properties of the underlying density and its independence structure simultaneously. Specifically, we obtain adaptive upper bounds over a scale of anisotropic Nikolskii classes when the smooth- ness is also measured with the Lp-norm. It is important to emphasize that the adaptation to unknown independence structure of the estimated density allows us to improve significantly the accuracy of estimation (curse of di- mensionality). The main technical tools used in our derivation are uniform bounds on the Lp-norms of empirical processes developed in Goldenshluger and Lepski [13]

Structural adaptive deconvolution under $${\mathbb{L}_p}$$ -losses

“…for n large enough. Finally, it is easily seen that we get the statement of Theorem 5 from ( 33) and (34). Similarly, the statement of Theorem 6 is obtained by applying Theorem 4.…”

“…. , d), it is asserted in Rebelles [34] that the answer is positive and that the proof of the corresponding minimax lower bound coincides with the one of Theorem 3 in Goldenshluger and Lepski [16], up to minor modifications to take into account the independence structure. For the deconvolution model, we conjecture that the answer is also positive if p ∈ [2, +∞) and that a minimax lower bound on N p,d (β, L, P) can be obtained, up to straightforward modifications, as in Lepski and Willer [25].…”

“…Indeed, if f has an independence structure P = ∅ and the smoothness parameter h is fixed and properly chosen then our procedure should choose the true partition P and the estimator f (h,P) should provide a better accuracy of estimation than the classical kernel-type estimator f h . This was illustrated by a short simulation study in Rebelles [34] for the density model (with direct observations), under the L 2 -loss.…”

“…First, we deal with optimal deconvolution of a multivariate density under L p and sup-norm losses. Next, as in Lepski [23] (under sup-norm loss) and in Rebelles [34] (under L p -losses) for the density model, we also take advantage of the fact that some coordinates of the X k 's may be independent from the others, but in a unified way.…”

Structural adaptation in the density model

“…The problems and models similar to those considered in the present paper were studied in Samarov and Tsybakov (2007), Amato et al (2010), Lepski (2013), Rebelles (2015a), Rebelles (2015b). We would like especially to mention the paper Samarov and Tsybakov (2004) where d-dimensional variant of our model was considered.…”