Density Deconvolution in a Non-standard Case of Heteroscedastic Noises
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Density Deconvolution in a Non‑standard Case of Heteroscedastic Noises Cao Xuan Phuong1 · Le Thi Hong Thuy2 Accepted: 6 September 2020 © Grace Scientific Publishing 2020
Abstract We study the density deconvolution problem with heteroscedastic noises whose densities are known exactly and Fourier-oscillating. Based on available data, we propose a nonparametric estimator depending on two regularization parameters. This estimator is shown to be consistency with respect to the mean integrated squared error. We then establish upper and lower bounds of the error over the Sobolev class of target density to give the minimax optimality of the estimator. In particular, this estimator is adaptive to the smoothness of the unknown target density. We finally demonstrate that the estimator achieves the minimax rates when the noise densities are supersmooth and ordinary smooth. Keywords Density deconvolution · Heteroscedastic noises · Fourier-oscillating density · Minimax rate Mathematics Subject Classification 62G07 · 62G20
1 Introduction In this paper, we consider the additive noise model
Yj = Xj + 𝜀j ,
(1.1)
j = 1, … , n,
where Yj ’s are independent observations, Xj ’s are independent and identically distributed (i.i.d.) random variables with unknown common density f, and 𝜀j ’s are independent random noises. Assume that all the variables X1 , ..., Xn , 𝜀1 , ..., 𝜀n are * Cao Xuan Phuong [email protected] Le Thi Hong Thuy [email protected] 1
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
2
Faculty of Basic Sciences, Van Lang University, Ho Chi Minh City, Vietnam
13
Vol.:(0123456789)
64
Page 2 of 17
Journal of Statistical Theory and Practice
(2020) 14:64
mutually independent. Each variable 𝜀j distributes with a known density gj , called a noise density, which depends on the observation number j. If g1 = ⋯ = gn , then the noises 𝜀j ’s are said to be homoscedastic; otherwise, the noises are heteroscedastic. Based on the observations Yj ’s and the densities gj’s, we aim to estimate the density f in a nonparametric strategy. This problem has been known as a density deconvolution problem. The density deconvolution problem is a kind of statistical inverse problems. It has been widely and intensively investigated in the past three decades, see, e.g., Meister [9] and references therein. Most of studies were concentrated on the setting of homoscedastic noises, e.g., Carroll and Hall [2], Devroye [6], Stefanski and Carroll [12], Fan [7], Pensky and Vidakovic [10], Comte et al. [4], Hall and Meister [8] and many others. However, the homoscedasticity of the noises is a rather strict assumption in many real applications. In practice, heteroscedasticity of noises arises once the observations Yj ’s are collected in different conditions. For instance, the observations might have been taken from different studies (see Walter [14]), or the measurement procedure might differ among all individuals (see Bennett and Franklin [1]). In spite of potential applicati
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