Fourier Methods for Estimating Mixing Densities and Distributions

Abstract

Let $X_1, X_2, \cdots$ be iid observations from a mixture density $f(x) = \int f(x \mid \theta)dG(\theta)$, where $f(x \mid \theta)$ is a known parametric family of density functions and $G$ is an unknown distribution function. This paper concerns estimating the mixing density $g = G'$ and the mixing distribution $G$. Fourier methods are used to derive kernel estimators, upper bounds for their rates of convergence and lower bounds for the optimal rate of convergence. Sufficient conditions are given under which the kernel estimators are asymptotically normal. Our estimators achieve the optimal rate of convergence $(\log n)^{-1/2}$ for the normal family and $(\log n)^{-1}$ for the Cauchy family.