On adaptive minimax density estimation on $$R^d$$

Abstract

We address the problem of adaptive minimax density estimation on R d with Lploss on the anisotropic Nikol'skii classes. We fully characterize behavior of the minimax risk for different relationships between regularity parameters and norm indexes in definitions of the functional class and of the risk. In particular, we show that there are four different regimes with respect to the behavior of the minimax risk. We develop a single estimator which is (nearly) optimal in order over the complete scale of the anisotropic Nikol'skii classes. Our estimation procedure is based on a data-driven selection of an estimator from a fixed family of kernel estimators.

Lower bounds in the convolution structure density model

“…The assertions of Theorem 1 in the case α ̸ = 0 are completely new. In the case α = 0, the lower bounds from Theorem 1 coincide with those found in Goldenshluger and Lepski (2014) when the tail or dense zone are considered. The result corresponding to the sparse zone even for α = 0 was not known.…”

“…Following the terminology used in Goldenshluger and Lepski (2014), we will call the set of parameters satisfying κ α (p) > pω(α) the tail zone, satisfying 0 < κ α (p) ≤ pω(α) the dense zone and satisfying κ α (p) ≤ 0 the sparse zone. In its turn, the latter zone is divided in two sub-domains: the sparse zone 1 corresponding to τ (p * ) > 0 and the sparse zone 2 corresponding to τ (p * ) ≤ 0.…”

“…In particular the consideration of compactly supported densities allows to eliminate this zone. The fact that the sparse zone is divided in two sub-domains was discovered in Goldenshluger and Lepski (2014) (bounded case) and in Lepski (2015) (unbounded case). This division can be informally viewed as some "degree of sparsity".…”

“…In fact, the main motivation lies in the calculation of the upper bounds of the minimax risk defined further. Indeed, in a separate work, the authors have constructed a performant estimation technique, along the lines of what Goldenshluger and Lepski (2014) developed in the direct case. However this estimator, like (to the authors knowledge) most of those available in the literature, exploit neither f ≥ 0 nor ∫ f = 1.…”

“…were obtained in Goldenshluger and Lepski (2014). The authors obtain the following lower and upper bounds for the minimax risk:…”