The Dagum family of isotropic correlation functions
By Christian Berg, Jorge Mateu, Emilio Porcu. 2007
61 citations recorded by the source corpus.
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.
Subfields: Geometry and Topology, Statistics and Probability
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- On the Schoenberg Transformations in Data Analysis: Theory and Illustrations (2010, background): “…Berg et al. 2008): where µ is a non-negative measure on (0 , ∞ ) such that There exists an important literature about Bernstein functions (see e.g. Berg et al. 2008; Schilling et al. 2010; and references therein), defined as the smooth non-negative functions whose first derivatives are completely…”