Bernoulli

  • 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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Abstract

A function ρ:[0, ∞)→(0, 1] is a completely monotonic function if and only if ρ(‖x2) is positive definite on ℝd 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.

Article information

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

Dates
First available in Project Euclid: 6 November 2008

Permanent link to this document
http://projecteuclid.org/euclid.bj/1225980574

Digital Object Identifier
doi:10.3150/08-BEJ139

Mathematical Reviews number (MathSciNet)
MR2543589

Zentralblatt MATH identifier
1158.60350

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

Citation

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. http://projecteuclid.org/euclid.bj/1225980574.


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