Estimation and Moment Recursion Relations for Multimodal Distributions of the Exponential Family

Multimodal generalizations of the normal, gamma, inverse gamma, and beta distributions are introduced within a unified framework. These multimodal distributions, belonging to the exponential family, require fewer parameters than corresponding mixture densities and have unique maximum likelihood estimators. Simple moment recursion relations, which make maximum likelihood estimation feasible, also yield easily computed estimators that themselves are shown to be consistent and asymptotically normal. Lastly, a statistic for bimodality, based on Cardan's discriminant for a cubic shape polynomial, is introduced.