Natural Exponential Families With Reduction Functions and Resolution of A Conjecture

Motivated by the need to estimate latent, low-dimensional linear structure and for dimension reduction in the first and/or second moments in high-dimensional data, we attempt to classify one-parameter natural exponential family (NEF) with the property that the variance of the random variable with such parametric distributions is the expectation of a deterministic function of the random variable only. We show that a subfamily of NEFs with power or polynomial variance functions has this property. This implies that, for data modelled by such distributions, estimating the latent, low-dimensional linear structure and a dimension reduction in their first two moments can be achieved in a fully data driven way. As a by-product of our investigation, we provide a proof of the conjecture raised by Bar-Lev, Bshouty, and Enis on classification of polynomials as variance functions for NEFs. The positive answer to this conjecture enlarges existing family of polynomials that are able to generate NEFs, and it helps prevent practitioners from choosing incompatible functions as variance functions for statistical modeling using NEFs.