[Paper Review] A Note on the Existence of the Multivariate Gamma Distribution

Thomas Royen|arXiv (Cornell University)|Jun 15, 2016

Bayesian Methods and Mixture Models11 references3 citations

TL;DR

This paper establishes the existence of the p-variate gamma distribution for all real degrees of freedom d > ⌊(p−1)/2⌋, without requiring restrictive assumptions on the associated covariance matrix. By deriving a relationship between central and non-central multivariate gamma distributions, the author proves existence for non-integer d greater than the floor of (p−1)/2, extending prior results that required integer d or specific covariance structures.

ABSTRACT

The p-variate gamma distribution in the sense of Krishnamoorthy and Parthasarathy exists for all positive integer degrees of freedom d and at least for all real values d > p-2, p > 1. For special structures of the "associated" covariance matrix it also exists for all positive d. In this paper a relation between central and non-central multivariate gamma distributions is shown, which implies the existence of the p-variate gamma distribution at least for all non-integer d greater than the integer part of (p-1)/2 without any additional assumptions for the associated covariance matrix.

Motivation & Objective

• To resolve the existence conditions of the multivariate gamma distribution beyond prior constraints.
• To extend the known existence domain of the p-variate gamma distribution to non-integer degrees of freedom.
• To eliminate the need for restrictive assumptions on the associated covariance matrix in the distribution's definition.
• To establish a theoretical foundation for broader applications of multivariate gamma distributions in statistics and probability.

Proposed method

• Derives a mathematical relationship between central and non-central multivariate gamma distributions.
• Uses this relationship to infer existence properties for non-integer degrees of freedom.
• Applies known results on the multivariate gamma distribution for integer d > p−2 and d ∈ ℕ.
• Leverages the connection between central and non-central forms to extend existence to d > ⌊(p−1)/2⌋.
• Analyzes the structure of the associated covariance matrix and shows that no additional assumptions are needed.
• Employs techniques from probability theory and multivariate distribution theory to prove existence under minimal conditions.

Experimental results

Research questions

• RQ1 For which values of d does the p-variate gamma distribution exist when d is not necessarily an integer?
• RQ2 Can the existence of the multivariate gamma distribution be established without imposing structural constraints on the associated covariance matrix?
• RQ3 What is the minimal lower bound on d for the existence of the p-variate gamma distribution in the general case?
• RQ4 How does the relationship between central and non-central multivariate gamma distributions inform the existence of the distribution?
• RQ5 Can prior existence results for integer d be extended to non-integer d using distributional relationships?

Key findings

• The p-variate gamma distribution exists for all real d > p−2 when p > 1.
• For special covariance matrix structures, existence holds for all d > 0.
• The distribution exists for all non-integer d > ⌊(p−1)/2⌋ without additional assumptions on the covariance matrix.
• The key result is the existence of the distribution for d > ⌊(p−1)/2⌋, which generalizes earlier results.
• The proof relies on a derived relationship between central and non-central multivariate gamma distributions.
• The result extends the domain of existence beyond integer degrees of freedom and removes restrictive assumptions on the covariance matrix.