The Deconvolution Problem

Abstract

We first consider the deconvolution problem in a model with non-negative random variables and disturbances with a decreasing density. Formally, let Z₁,..., Z _n be a sample from a distribution function H with density

$$h(z) = \int {g(z - x)d{F_0}(x),z \in R} $$

(2.1)

where g is a decreasing density on [0, ∞), and F₀ an unknown distribution function, concentrated on [0, ∞). For example, g could be the exponential density

$$g(x) = {e^{ - x}}{1_{[0,\infty )}}(x),x \in R$$

or the Uniform (0,1) density

$$g(x) = {1_{[0,1)}}(x),x \in R$$

An NPMLE of F₀ is a distribution function, maximizing

$$\psi (F) = \int {\log \left\{ {\int g (z - x)dF(x)} \right\}d{H_n}(z)} $$

(2.2)

as a function of F, where H _n is the empirical distribution function of the sample Z₁,..., Z _n .