### Abstract

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 language | English |
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Pages (from-to) | 2285-2304 |

Number of pages | 20 |

Journal | Communications in Statistics - Theory and Methods |

Volume | 32 |

Issue number | 12 |

DOIs | |

Publication status | Published - 2003 Dec 1 |

Externally published | Yes |

### Keywords

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

### ASJC Scopus subject areas

- Statistics and Probability