The Dagum family of isotropic correlation functions

Details The Dagum family of isotropic correlation functions The Dagum family of isotropic correlation functions

2007-05-03

arxiv.org/abs/0705.0456v2

Bernoulli 2008, Vol. 14, No. 4, 1134-1149

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.3150/08-BEJ139 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/08-BEJ139

A function $\rho:[0,\infty)\to(0,1]$ is a completely monotonic function if and only if $\rho(\Vert\mathbf{x}\Vert^2)$ is positive definite on $\mathbb{R}^d$ for all $d$ and thus it represents the correlation function of a weakly stationary and isotropic Gaussian random field. Radial positive definite functions are also of importance as they represent characteristic functions of spherically symmetric probability distributions. In this paper, we analyze the function \[\rho(\beta ,\gamma)(x)=1-\biggl(\frac{x^{\beta}}{1+x^{\beta}}\biggr )^{\gamma},\qquad x\ge 0, \beta,\gamma>0,\] called the Dagum function, and show those ranges for which this function is completely monotonic, that is, positive definite, on any $d$-dimensional Euclidean space. Important relations arise with other families of completely monotonic and logarithmically completely monotonic functions.

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