The Dagum family of isotropic correlation functions

Christian Berg, Jorge Mateu, Emilio Porcu

Research output: Contribution to journalArticlepeer-review

41 Scopus citations

Abstract

A function ρ: [0, ∞) → (0, 1] is a completely monotonic function if and only if ρ(∥x∥2) 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 ρ(β, γ)(x)=1-(xβ/1+xβ)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.

Original languageBritish English
Pages (from-to)1134-1149
Number of pages16
JournalBernoulli
Volume14
Issue number4
DOIs
StatePublished - Nov 2008

Keywords

  • Bernstein function
  • Completely monotonic function
  • Dagum family
  • Isotropy
  • Logarithmically completely monotonic function
  • Stieltjes transform

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