A Functional EM Algorithm for Mixing Density Estimation via Nonparameteric Penalized Likelihood Maximization

When the true mixing distribution is known to be continuous, the nonparametric maximum likelihood estimate of the mixing distribution cannot provide a satisfying answer due to its degeneracy. The estimation of mixing densities is an ill-posed indirect problem. In this article, we propose to estimate the mixing density by maximizing a penalized likelihood and call the resulting estimate the nonparametric maximum penalized likelihood estimate (NPMPLE). Using theory and methods from the calculus of variations and differential equations, a new functional EM algorithm is derived for computing the NPMPLE of the density. In the algorithm, maximizers in M-steps are found by solving an ordinary differential equation with boundary conditions numerically. Simulation studies show the algorithm outperforms other existing methods such as the popular EMS algorithm and the kernel method. Some theoretical properties of the NPMPLE and the algorithm are also given in the article.