Minimax bounds for Besov classes in density estimation

Abstract

We study the problem of density estimation on [0, 1] under L p norm. We carry out a new piecewise polynomial estimator and prove that it is simultaneously (near)-minimax over a very wide range of Besov classes B α π,∞ (R). In particular, we may deal with unbounded densities and shed light on the minimax rates of convergence when π < p and α ∈ (1/π − 1/p, 1/π].

Density estimation under local differential privacy and Hellinger loss

“…This point is also true for our procedure. For more information on these computational aspects, we refer to [BSR04,AL11,Sar14,Sar21b]. The theoretical results of these methods are, however, substantially different from ours.…”

“…How to choose this partition m remains to be decided. The solution we propose below is to use a Lespki-type procedure [Lep92] modified as in [Sar14,Sar21b] for computational reasons.…”

“…Even in the direct case, it is only very recently that results have been established. Actually, the estimation rate of a bounded or unbounded density when β ≤ 1/p may not be the same, see [Bir14,Sar21b] for more details. It turns out that a similar phenomenon occurs in the private setup.…”

“…where c depends on β, p only, see Lemma 5 of [Sar21b] for instance. Moreover, for all x, x + h ∈ [0, 1],…”

“…The two following points are results from the approximation theory that can be found in the literature. We refer, for instance, to Corollary 3.3 of [DY90], Lemma 12 of [BBM99], Theorem 2 of [Aka12] or Proposition 5 of [Sar21b].…”