Statistics Papers

Document Type

Journal Article

Date of this Version

2014

Publication Source

The American Statistician

Volume

68

Issue

3

Start Page

183

Last Page

187

DOI

10.1080/00031305.2014.909741

Abstract

The bivariate normal density with unit variance and correlation ρ is well known. We show that by integrating out ρ, the result is a function of the maximum norm. The Bayesian interpretation of this result is that if we put a uniform prior over ρ, 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 ρ is its special case in two dimensions.

Copyright/Permission Statement

This is an Accepted Manuscript of an article published by Taylor & Francis in The American Statistician on 14 Apr 2014, available online: http://wwww.tandfonline.com/10.1080/00031305.2014.909741.

Keywords

bivariate normal mixture, khintchine mixture, uniform prior over correlation

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Date Posted: 27 November 2017