A Multivariate Extension of Inverse Gaussian Distribution Derived from Inverse Relationship

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We propose a new multivariate extension of the inverse Gaussian distribution derived from a certain multivariate inverse relationship. First we define a multivariate extension of the inverse relationship between two sets of multivariate distributions, then define a reduced inverse relationship between two multivariate distributions. We derive the multivariate continuous distribution that has the reduced multivariate inverse relationship with a multivariate normal distribution and call it a multivariate inverse Gaussian distribution. This distribution is also characterized as the distribution of the location of a multivariate Brownian motion at some stopping time. The marginal distribution in one direction is the inverse Gaussian distribution, and the conditional distribution in the space perpendicular to this direction is a multivariate normal distribution. Mean, variance, and higher order cumulants are derived from the multivariate inverse relationship with a multivariate normal distribution. Other properties such as reproductivity and infinite divisibility are also given.

Original languageEnglish
Pages (from-to)2285-2304
Number of pages20
JournalCommunications in Statistics - Theory and Methods
Issue number12
Publication statusPublished - 2003 Dec 1
Externally publishedYes



  • Cumulant generating function
  • Inverse relationship
  • Multivariate Brownian motion
  • Multivariate normal distribution

ASJC Scopus subject areas

  • Statistics and Probability

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