Moment varieties of the inverse Gaussian and gamma distributions are nondefective

TL;DR

This paper proves that the parameters of mixtures of inverse Gaussian and gamma distributions can be algebraically and rationally identified from a finite number of moments, using advanced algebraic geometry techniques.

Contribution

It establishes the algebraic and rational identifiability of mixture parameters from moments, extending the understanding of moment varieties for these distributions.

Findings

Parameters are algebraically identifiable from the first 3k-1 moments.

Parameters are rationally identifiable from the first 3k+2 moments.

The proofs utilize classification of defective surfaces and secant variety theory.

Abstract

We show that the parameters of a $k$-mixture of inverse Gaussian or gamma distributions are algebraically identifiable from the first $3k-1$ moments, and rationally identifiable from the first $3k+2$ moments. Our proofs are based on Terracini's classification of defective surfaces, careful analysis of the intersection theory of moment varieties, and a recent result on sufficient conditions for rational identifiability of secant varieties by Massarenti--Mella.