# Relative variation indexes for multivariate continuous distributions on $[0,\infty)^k$ and extensions

## TL;DR

This paper introduces new multivariate variation indexes for non-negative distributions, useful for comparing and discriminating between different models based on their deviation from a reference distribution.

## Contribution

The paper proposes novel scalar indexes based on quadratic forms of mean and covariance, extending relative variation measures to multivariate continuous distributions on $[0,
olinebreak ext{ extasciitilde}]
olinebreak^k$.

## Key findings

- Indexes effectively discriminate between positive distributions.
- Asymptotic properties of indexes are established.
- Numerical examples demonstrate practical applications.

## Abstract

We introduce some new indexes to measure the departure of any multivariate continuous distribution on non-negative orthant from a given reference one such the uncorrelated exponential model, similar to the relative Fisher dispersion indexes of multivariate count models. The proposed multivariate variation indexes are scalar quantities, defined as ratios of two quadratic forms of the mean vector and the covariance matrix. They can be used to discriminate between continuous positive distributions. Generalized and multiple marginal variation indexes with and without correlation structure, respectively, and their relative extensions are discussed. The asymptotic behavior and other properties are studied. Illustrative examples and numerical applications are analyzed under several scenarios, leading to appropriate choices of multivariate models. Some concluding remarks and possible extensions are made.