Tikhonov’s Regularization to the Deconvolution Problem

Abstract

We are interested in estimating a density function f of i.i.d. random variables X₁, …, X_n from the model Y_j = X_j + Z_j, where Z_j are unobserved error random variables, distributed with the density function g and independent of X_j. This problem is known as the deconvolution problem in nonparametric statistics. The most popular method of solving the problem is the kernel one in which, we assume g^ft(t) ≠ 0, for all , where g^ft(t) is the Fourier transform of g. The more general case in which g^ft(t) may have real zeros has not been considered much. In this article, we will consider this case. By estimating the Lebesgue measure of the low level sets of g^ft and combining with the Tikhonov regularization method, we give an approximation f_n to the density function f and evaluate the rate of convergence of . A lower bound for this quantity is also provided.

Keywords:

• Deconvolution
• Tikhonov’s regularization
• Fourier transform

Mathematics Subject Classification:

• 62G05
• 62G07

Additional information

Funding

This article is supported by National Foundation of Scientific and Technology Development (NAFOSTED) Project 101.01-2012.07.