### 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 |

Externally published | Yes |

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### Keywords

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

### ASJC Scopus subject areas

- Statistics and Probability
- Safety, Risk, Reliability and Quality

### Cite this

**A Multivariate Extension of Inverse Gaussian Distribution Derived from Inverse Relationship.** / Minami, Mihoko.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - A Multivariate Extension of Inverse Gaussian Distribution Derived from Inverse Relationship

AU - Minami, Mihoko

PY - 2003/12

Y1 - 2003/12

N2 - 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.

AB - 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.

KW - Cumulant generating function

KW - Inverse relationship

KW - Multivariate Brownian motion

KW - Multivariate normal distribution

UR - http://www.scopus.com/inward/record.url?scp=0344861863&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0344861863&partnerID=8YFLogxK

U2 - 10.1081/STA-120025379

DO - 10.1081/STA-120025379

M3 - Article

AN - SCOPUS:0344861863

VL - 32

SP - 2285

EP - 2304

JO - Communications in Statistics - Theory and Methods

JF - Communications in Statistics - Theory and Methods

SN - 0361-0926

IS - 12

ER -