Lower bounds in the convolution structure density model

Abstract

Abstract:The aim of the paper is to establish asymptotic lower bounds for the minimax risk in two generalized forms of the density deconvolution problem. The observation consists of an independent and identically distributed (i.i.d.) sample of n random vectors in R d . Their common probability distribution function p can be written aswhere f is the unknown function to be estimated, g is a known function, α is a known proportion, and ⋆ denotes the convolution product. The bounds on the risk are established in a very general minimax setting and for moderately ill posed convolutions. Our results show notably that neither the ill-posedness nor the proportion α play any role in the bounds whenever α ∈ [0, 1), and that a particular inconsistency zone appears for some values of the parameters. Moreover, we introduce an additional boundedness condition on f and we show that the inconsistency zone then disappears.AMS 2000 subject classifications: 62G05, 62G20.

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

“…Unfortunately, if p ∈ (1, 2), our estimator does not achieve the minimax lower bound on N p,d (β, L) obtained in Lepski and Willer [25] under the L p -loss. We conclude that either our estimator is not minimax on N p,d (β, L) or the lower bound in Lepski and Willer [25] is not the minimax rate of convergence on the latter functional class.…”

“…Finally, we conjecture that φ n,∞ (β, r, P) is the minimax rate of convergence on N r,d (β, L, P) when 1 − d j=1 1 β j r j > 0 and that a proof of the corresponding lower bound can be obtained by a minor modification of that in Lepski and Willer [25] to take into account the possible independence structure of the underlying density.…”

“…Here, the optimality is a direct consequence of minimax lower bounds recently obtained by Lepski and Willer [25]. As usually, these lower bounds hold under additional assumptions on the common density of the errors, see Section 2.4.…”

“…Here we have used the product structure of the kernel K and the Fubini theorem. Thus, in view of the triangle inequality, (25), (27) and the trivial inequality…”

Oracle inequalities and adaptive estimation in the convolution structure density model

“…Introduction. In the present paper we will investigate the following observation scheme introduced in Lepski and Willer (2017). Suppose that we observe i.i.d.…”