Deriving Mixture Distributions Using Moment Generating Functions: A Hierarchical Model Approach
Subhash Bagui *
Department of Mathematics and Statistics, The University of West Florida, Pensacola, FL, USA.
Jia Liu
Department of Mathematics and Statistics, The University of West Florida, Pensacola, FL, USA.
Shen Zhang
Meta, USA.
*Author to whom correspondence should be addressed.
Abstract
Mixture distributions are important because they model data from multiple underlying subpopulations, allowing us to capture heterogeneity that a single distribution can’t explain. Generally, mixture distributions arise as marginal distributions of hierarchical mixture models. In this chapter, we use moment-generating functions (mgfs) to derive the densities of mixture distributions from hierarchical models. When the mgf of a mixture distribution doesn’t exist, the approach can be extended to characteristic functions to derive the mixture density. This chapter uses a result from Villa and Escobar (2006). The present work complements Villa and Escobar’s (2006) article with many new examples.
Keywords: Mixture distributions, moment generating functions, characteristic functions, hierarchical models, over-dispersed models