Nonparametric Estimation of a Mixing Density via the Kernel Method

Abstract

Editor's note: This manuscript was first submitted in May 1996, and a first review with a request for revision was completed in September 1996. Costas Goutis lost his life in a scuba diving accident in July 1996. The associate editor, Jianqing Fan, graciously offered to carry out the required revisions.

This article presents a method for estimating the latent distribution of a mixture model. The method is motivated by the standard kernel density estimation, but instead of using an estimate based on the unobserved latent variables, it takes the expectation with respect to their distribution conditional on the data. The resulting estimator is continuous and hence appropriate when there is a strong belief in the continuity of the mixing distribution. An asymptotic justification is presented, and the associated computational problems are discussed. The method is illustrated by an example of fission track analysis in which the density of the age of crystals is estimated.

Key Words:

• Continuous mixtures
• Cross-validation
• Fission track analysis
• Kernel density estimation.