The Dagum family of isotropic correlation functions

2007/05/31 by Christian Berg, Jorge Mateu, Emilio Porcu · 2 citations
Mathematics · #Geometry and complex manifolds #Random Matrices and Applications #Statistical Distribution Estimation and Applications #math.ST #stat.TH

paper · pdf · doi:10.3150/08-bej139

published as Bernoulli 2008, Vol. 14, No. 4, 1134-1149 · Published in at http://dx.doi.org/10.3150/08-BEJ139 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

openalex publication_date 2008/11/01 · arxiv created 2008/11/17 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

A function ρ:[0, ∞)→(0, 1] is a completely monotonic function if and only if ρ(‖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 ρ(β ,γ)(x)=1-\biggl(\fracx^β1+x^β\biggr)^γ, x≥ 0, β,γ>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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