V. Seshadri

Multivariate distributions with generalized inverse gaussian marginals, and associated poisson mixtures

Several types of multivariate extensions of the inverse Gaussian (IG) distribution and the reciprocal inverse Gaussian (RIG) distribution are proposed. Some of these types are obtained as random-additive-effect models by means of well-known convolution properties of the IG and RIG distributions, and they have one-dimensional IG or RIG marginals. They are used to define a flexible class of multivariate Poisson mixtures.

The inverse gaussian distribution: Some properties and characterizations†

This article presents some structural properties of the inverse Gaussian distribution, together with several new characterizations based on constancy of regression of suitable functions on the sum of n independent identically distributed random variables. A decomposition of the statistic λσ (X−1i−X−1) into n - 1 independent chi-squared random variables, each with one degree of freedom, is given when n is of the form 2r.

A family of distributions related to the McCullagh family

A family of distributions on (-1, 1) is obtained from the hypergeometric functions which bears some resemblance to a similar family introduced recently by McCullagh. This family shares several properties with the McCullagh family. In particular the family possesses a pivot which reduces it to the symmetric beta density. Relations to proper dispersion models and the exactness of the p*-formula are also considered.

The limit behavior of an interval splitting scheme

We split [0,1] in a uniform manner, take the largest of the two intervals thus obtained, split this interval again uniformly, and continue in this fashion ad infinitum. We show that the extremes of this interval converge almost surely to a beta (2,2) random variable.

General exponential models on the unit simplex and related multivariate inverse Gaussian distributions

Barndorff-Nielsen and Jorgensen (1989) have introduced some parametric models on the unit simplex. The distributions associated with these models have been obtained by conditioning on the sum of d independent generalized inverse Gaussian random variables. We use a constructive approach to derive some of these models by first mapping the inverse Gaussian law on (0, 1) and formally extending it on the unit simplex. This technique is then applied to a mixture-inverse Gaussian distribution studied recently by Jorgensen, Seshadri and Whitmore (1991). The distributions are then retransformed to yield two versions of a multidimensional inverse Gaussian distribution.

A U-statistic and estimation for the inverse Gaussian distribution

Unbiased estimation of the square of the coefficient of variation in the inverse Gaussian distribution is considered. It is shown that the estimator is indeed a U-statistic. The exact distribution of the estimator is also derived.

Haight's distributions as a natural exponential family

In an index to the distributions of Mathematical Statistics published in 1961, Frank A. Haight considers, without giving any references, the following distribution: for 0

Martingales Defined by Reciprocals of Sums and Related Characterizations

Abstract We prove that linearly transformed inverses of cumulative sums form backward martingales for gamma, inverse Gaussian, and Kendall and Borel–Tanner sequences of independent, identically distributed random variables. Conversely, a characterization of the family of these four distributions by linearity of regression of inverses of sums is obtained. The results in both directions are derived via the technique of variance functions of natural exponential families.

On a property of strongly reproductive exponential families on

Strongly reproductive exponential models with affine dual foliations are known to allow of a decomposition analogous to the standard decomposition theorem for Chi-squared distributed quadratic forms in normal variates. It is shown that when the components are identically distributed, then necessarily each component follows the gamma law.

On a Conjecture concerning Inverse Gaussian Regression

The scaled deviance of an inverse Gaussian sample of size n can be expressed as a sum of (n 1) independent chi-squared variates, a result paralleling that for the Gaussian case. Whitmore (1985) conjectured that, just as for the Gaussian model, when a multiple regression model is assumed for inverse Gaussian data, the deviance for multiple regression can be decomposed into mutually independent (regression and error) chi-squared distributed components. This note considers a multiple regression model for which the conjecture is shown to be false.