Uniform Correlation Mixture of Bivariate Normal Distributions and Hypercubically-contoured Densities That Are Marginally Normal

TL;DR

This paper explores a family of multivariate distributions formed by integrating bivariate normal densities over all correlation values, resulting in densities with hypercubically-shaped contours and a connection to the Khintchine mixture method.

Contribution

It introduces a new class of multivariate densities that are marginally normal and depend on the maximum norm, generalizing the bivariate case and linking to existing mixture methods.

Findings

The marginal bivariate density with integrated correlation has square-shaped isodensity contours.

Higher-dimensional generalizations of these densities are constructed using the Khintchine mixture method.

For each dimension, a unique multivariate density exists that is a differentiable function of the maximum norm.

Abstract

The bivariate normal density with unit variance and correlation $\rho$ is well-known. We show that by integrating out $\rho$, the result is a function of the maximum norm. The Bayesian interpretation of this result is that if we put a uniform prior over $\rho$, then the marginal bivariate density depends only on the maximal magnitude of the variables. The square-shaped isodensity contour of this resulting marginal bivariate density can also be regarded as the equally-weighted mixture of bivariate normal distributions over all possible correlation coefficients. This density links to the Khintchine mixture method of generating random variables. We use this method to construct the higher dimensional generalizations of this distribution. We further show that for each dimension, there is a unique multivariate density that is a differentiable function of the maximum norm and is marginally normal, and the bivariate density from the integral over $\rho$ is its special case in two dimensions.