Point-Wise Wavelet Estimation in the Convolution Structure Density Model
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(2020) 26:81
Point-Wise Wavelet Estimation in the Convolution Structure Density Model Youming Liu1 · Cong Wu1 Received: 31 October 2019 / Revised: 21 July 2020 / Accepted: 2 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract By using a kernel method, Lepski and Willer establish adaptive and optimal L p risk estimations in the convolution structure density model in 2017 and 2019. They assume their density functions to be in a Nikol’skii space. Motivated by their work, we first use a linear wavelet estimator to obtain a point-wise optimal estimation in the same model. We allow our densities to be in a local and anisotropic Hölder space. Then a data driven method is used to obtain an adaptive and near-optimal estimation. Finally, we show the logarithmic factor necessary to get the adaptivity. Keywords Density estimation · Generalized deconvolution model · Point-wise risk · Optimality · Adaptivity · Wavelet · Anisotropic Hölder space Mathematics Subject Classification 42C40 · 62G07 · 62G20
1 Introduction The deconvolution estimation is an important topic in statistics. In this paper, we consider the generalized deconvolution model introduced in [21,22]. Let (, F, P) be a probability space and Z 1 , Z 2 , . . . , Z n be independent and identically distributed (i.i.d.) random variables of Z = X + εY ,
(1.1)
Communicated by Stephane Jaffard.
B
Cong Wu [email protected] Youming Liu [email protected]
1
Department of Applied Mathematics, Beijing University of Technology, Beijing 100124, P. R. China 0123456789().: V,-vol
81
Page 2 of 28
Journal of Fourier Analysis and Applications
(2020) 26:81
where X stands for a real-valued random variable with unknown probability density f on Rd , Y denotes an independent random noise (error) with the probability density g and ε ∈ {0, 1} Bernoulli random variable with P{ε = 1} = α, α ∈ [0, 1]. The purpose is to estimate f by the observed data Z 1 , Z 2 , . . . , Z n in some sense. When α = 1, (1.1) reduces to the classical deconvolution model, while α = 0 corresponds to the traditional density estimation. Clearly, the density h of Z in (1.1) satisfies h = (1 − α) f + α f ∗ g,
(1.2)
because P{Z < t} = P{ε = 0}P{X + εY |ε=0 < t} + P{ε = 1}P{X + εY |ε=1 < t} = (1 − α)P{X < t} + α P{X + Y < t}. Furthermore, f
ft
ft (t) = [(1 − α) + αg f t (t)]−1 h f t (t) := G −1 α (t)h (t),
when the function G α (t) = 1 − α + αg f t (t) has nonzeros on Rd , where f
ft
f
ft
is the Fourier transform of f ∈ L 1 (Rd ) defined by
(t) :=
Rd
f (x)e−it x d x.
Under some mild assumptions on G α (t), Lepski and Willer [21] establish an asymptotic lower bound of L p risk for model (1.1). Recently, they provide an adaptive and optimal estimate of L p risk over an anisotropic Nikol’skii space by using kernel method in [22]. When α = 0, References [9] and [17] deal with linear wavelet estimations; Nonlinear wavelet estimations are studied for adaptivity [6,8,25]. Kernel estimation with selection rule can be found in References [12,13] and [20]. For
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