Ridge-Parameter Regularization to Deconvolution Problem with Unknown Error Distribution

• Published in: Vietnam journal of mathematics Vol. 43; no. 2; pp. 239 - 256
• Main Authors: Trong, Dang Duc, Phuong, Cao Xuan
• Format: Journal Article
• Language: English
• Published: Singapore Springer Singapore 01.06.2015
• Subjects: Mathematics; Mathematics and Statistics; Mean integrated squared error; Deconvolution; 62G07; Estimator; Fourier transform; Density function; 62F12
• ISSN: 2305-221X; 2305-2228
• DOI: 10.1007/s10013-015-0119-1

Our aim in this article is to estimate a density function f of i.i.d. random variables X 1 , … , X n from a noise model Y j = X j + Z j , j = 1, 2, … , n . Here, ( Z j ) 1≤ j ≤ n is independent of ( X j ) 1≤ j ≤ n and is a finite sequence of i.i.d. noise random variables distributed with an unknown density function g . This problem is known as the deconvolution problem in nonparametric statistics. The general case in which the error density function g is unknown and its Fourier transform g ft can vanish on a subset of ℝ has still not been considered much. In the present article, we consider this case. Using direct i.i.d. data Z 1 ′ , … , Z m ′ which are collected in separated independent experiments, we propose an estimator ĝ to the unknown density function g . After that, applying a ridge-parameter regularization method and an estimation of the Lebesgue measure of low level sets of g ft , we give an estimator f ̂ to the target density function f and evaluate therateof convergence of the quantity 𝔼 ∥ f ̂ − f ∥ 2 2 .