• Bernoulli
  • Volume 14, Number 4 (2008), 1134-1149.

The Dagum family of isotropic correlation functions

Christian Berg, Jorge Mateu, and Emilio Porcu

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A function $ρ:[0, ∞)→(0, 1]$ is a completely monotonic function if and only if $ρ(‖\mathbf{x}‖^2)$ is positive definite on $ℝ^d$ 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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Bernoulli, Volume 14, Number 4 (2008), 1134-1149.

First available in Project Euclid: 6 November 2008

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Bernstein function completely monotonic function Dagum family isotropy logarithmically completely monotonic function Stieltjes transform


Berg, Christian; Mateu, Jorge; Porcu, Emilio. The Dagum family of isotropic correlation functions. Bernoulli 14 (2008), no. 4, 1134--1149. doi:10.3150/08-BEJ139.

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