Consistent estimation of finite mixtures: An application to latent group panel structures

In this presentation, I show that maximizing the likelihood of a mixture of a finite number of parametric densities leads to inconsistent estimates under weak regularity conditions. The size of the asymptotic bias is positively correlated with the overall degree of overlap between the densities within the mixture. In contrast, I show that slight modifications in the classification expectation-maximization (CEM) algorithm—the likelihood generalization of the K-means algorithm—produce consistent estimates of all parameters in the mixture, and I derive the asymptotic distribution of the proposed estimation procedure. I confirm the inconsistency of MLE procedures, such as the expectation-maximization (EM) algorithm, using numerical experiments with simple Gaussian mixture models. Simulation results show that the proposed estimation strategy generally outperforms the EM algorithm when estimating latent group panel structures with unrestricted group membership across units and over time. I also compare the finite-sample performance of each estimation strategy using a mixture of two-part models to predict individual healthcare expenditures from health administrative data. Estimation results show that the proposed consistent CEM approach leads to smaller prediction errors than models estimated with the EM algorithm, with a reduction of more than 40% in the out-of-sample prediction error compared with the standard, single-component, two-part model. The proposed estimation procedure thus represents a useful tool when both homogeneity of the parameters and constant group membership are assumed not to hold in panel-data analysis.