Lower bounds in the convolution structure density model
| Autor: | Thomas Willer, Oleg Lepski |
|---|---|
| Přispěvatelé: | Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), ANR-11-IDEX-0001,Amidex,INITIATIVE D'EXCELLENCE AIX MARSEILLE UNIVERSITE(2011) |
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
Statistics and Probability
Independent and identically distributed random variables Nikol'skii spaces Star (game theory) generalized deconvolution model 010102 general mathematics adaptive estimation Nikol’skii spaces Probability density function Density estimation Function (mathematics) $\mathbb{L}_{p}$-risk Minimax 01 natural sciences minimax rates Lp–risk Convolution Combinatorics 010104 statistics & probability [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST] Product (mathematics) density estimation 0101 mathematics Mathematics |
| Zdroj: | Bernoulli Bernoulli, 2017, 23 (2), pp.884-926. ⟨10.3150/15-BEJ763⟩ Bernoulli, Bernoulli Society for Mathematical Statistics and Probability, 2017, 23 (2), pp.884-926. ⟨10.3150/15-BEJ763⟩ Bernoulli 23, no. 2 (2017), 884-926 |
| ISSN: | 1350-7265 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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