Elliptical Distributions with Conditional Second Moments Linear in the Empirical Second Moment of the Conditioning Vector: Multivariate Pearson Type VII Family and Its Applications

• Activity Number: 43
• Type: Contributed
• Date/Time: Sunday, July 31, 2011 : 2:00 PM to 3:50 PM
• Sponsor: SSC
• Author(s): Weiyu Qiu*+ and Yutaka Yasui
• Companies: University of Alberta and University of Alberta
• Keywords: conditional moments ; multivariate Pearson Type VII family ; multivariate t distribution ; multivariate regression ; coverage probability

Abstract

Multivariate normal (MVN) distributions are commonly used to model multivariate dependency. Concerns have been raised since they may not represent real underlying stochastic processes as their conditional second moments do not depend on the conditioning vectors. To relax this property, we consider the elliptical family (a family of multivariate distributions whose regressions are linear) and seek a subfamily in which every conditional second moment is linear in the empirical second moment of the conditioning vector. We show that this property holds if and only if the distribution belongs to the symmetric multivariate Pearson Type VII (MP VII) family. A proof shows that if the underlying distribution is multivariate t (an MP VII distribution), the coverage probability of MVN-based conditional prediction intervals is different from their nominal level. This was confirmed with a simulation and demonstrated with real multivariate data that appear to follow an MP VII distribution, with a graphical approach for assessing MP VII vs. MVN for the distribution of a given data set.