Ridge-Parameter Regularization to Deconvolution Problem with Unknown Error Distribution

Abstract

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 $Z^{\prime }_{1},\ldots , Z^{\prime }_{m}$ which are collected in separated independent experiments, we propose an estimator $\hat g$ 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 $\hat f$ to the target density function f and evaluate therateof convergence of the quantity $\mathbb {E}\|\hat f - f\|_{2}^{2}$ .