Maximum Smoothed Likelihood Component Density Estimation in Mixture Models with Known Mixing Proportions

TL;DR

This paper introduces a maximum smoothed likelihood approach for estimating component densities in mixture models with known, possibly varying, mixing proportions, providing theoretical guarantees and demonstrating improved efficiency over existing methods.

Contribution

It develops a novel smoothed likelihood estimation method with proven convergence and asymptotic properties for mixture models with known mixing proportions.

Findings

The proposed method converges to the maximum smoothed likelihood estimates.

It achieves better efficiency than existing methods in simulations.

The approach is validated with a real data example.

Abstract

In this paper, we propose a maximum smoothed likelihood method to estimate the component density functions of mixture models, in which the mixing proportions are known and may differ among observations. The proposed estimates maximize a smoothed log likelihood function and inherit all the important properties of probability density functions. A majorization-minimization algorithm is suggested to compute the proposed estimates numerically. In theory, we show that starting from any initial value, this algorithm increases the smoothed likelihood function and further leads to estimates that maximize the smoothed likelihood function. This indicates the convergence of the algorithm. Furthermore, we theoretically establish the asymptotic convergence rate of our proposed estimators. An adaptive procedure is suggested to choose the bandwidths in our estimation procedure. Simulation studies show that the proposed method is more efficient than the existing method in terms of integrated squared errors. A real data example is further analyzed.