Simple Cubic Variance Functions on $\R^n$, Part one

TL;DR

This paper extends the classification of natural exponential families with polynomial variance functions from univariate to multivariate cases in ^n, focusing on degree three variance functions and their group actions.

Contribution

It generalizes the classification of exponential families with polynomial variance functions to multivariate ^n, including degree three functions, using group actions and multivariate calculus.

Findings

Extended classification to ^n for degree three variance functions

Identified group actions related to the classification

Provided a framework for multivariate variance function analysis

Abstract

The classification of natural exponential families started with the paper \cite {Morri} where Carl Morris unifies six very familiar families by the fact that their variance functions are polynomials of degree less or equal to two. Extension of this classification to $\R^n$ and to degree three is the subject of this paper. Keywords: Actions of the group $GL(n+1,\R)$, classification of natural exponential families, multivariate Lagrange formula. variance functions.