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May 2017 Lower bounds in the convolution structure density model
O.V. Lepski, T. Willer
Bernoulli 23(2): 884-926 (May 2017). DOI: 10.3150/15-BEJ763


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 $\mathbb{R}^{d}$. Their common probability distribution function $\mathfrak{p}$ can be written as $\mathfrak{p}=(1-\alpha)f+\alpha[f\star g]$, where $f$ is the unknown function to be estimated, $g$ is a known function, $\alpha$ is a known proportion, and $\star$ 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 $\alpha$ play any role in the bounds whenever $\alpha\in[0,1)$, 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.


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O.V. Lepski. T. Willer. "Lower bounds in the convolution structure density model." Bernoulli 23 (2) 884 - 926, May 2017.


Received: 1 January 2015; Revised: 1 May 2015; Published: May 2017
First available in Project Euclid: 4 February 2017

zbMATH: 1380.62208
MathSciNet: MR3606754
Digital Object Identifier: 10.3150/15-BEJ763

Keywords: $\mathbb{L}_{p}$-risk , adaptive estimation , Density estimation , generalized deconvolution model , Minimax rates , Nikol’skii spaces

Rights: Copyright © 2017 Bernoulli Society for Mathematical Statistics and Probability


Vol.23 • No. 2 • May 2017
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