Deconvolution estimation of mixture distributions with boundaries

Abstract

In this paper, motivated by an important problem in evolutionary biology, we develop two sieve type estimators for distributions that are mixtures of a finite number of discrete atoms and continuous distributions under the framework of measurement error models. While there is a large literature on deconvolution problems, only two articles have previously addressed the problem taken up in our article, and they use relatively standard Fourier deconvolution. As a result the estimators suggested in those two articles are degraded seriously by boundary effects and negativity. A major contribution of our article is correct handling of boundary effects; our method is asymptotically unbiased at the boundaries, and also is guaranteed to be nonnegative. We use roughness penalization to improve the smoothness of the resulting estimator and reduce the estimation variance. We illustrate the performance of the proposed estimators via our real driving application in evolutionary biology and two simulation studies. Furthermore, we establish asymptotic properties of the proposed estimators.

Least squares sieve estimation of mixture distributions with boundary effects

“…Lee et al (2013) used maximum likelihood to estimate these parameters. Below, we propose two alternative least-squares (LS) methods to estimate them.…”

“…To reduce this roughness, we propose minimizing the penalized distance between the two distributions, similar to the suggestion of Lee et al (2013).…”

“…Compared with the other studies, Lee et al (2013) covers more general cases of measurement error models that have two features: (1) discrete and continuous mixtures and (2) non-smooth boundaries. First they approximate the distribution of X using discretization, which gives a sieve of the distribution family.…”

“…However, the ML method of Lee et al (2013) involves long computation time and is not robust against misspecified error distribution. In this study, we propose alternative least squares (LS) sieve estimators based on the cumulative distribution function and characteristic function, instead of maximum likelihood.…”

“…This research is motivated by the maximum likelihood (ML) sieve estimator developed in Lee et al (2013). We obtain two types of LS sieve estimators through minimizing the distance between the empirical distribution/characteristic functions and the model distribution/characteristic functions.…”