The Dagum family of isotropic correlation functions

TL;DR

This paper investigates the conditions under which the Dagum family of functions serve as valid isotropic correlation functions and explores their connections with other monotonic function families.

Contribution

It characterizes the parameter ranges for the Dagum functions to be completely monotonic and positive definite across all dimensions, linking them to other monotonic functions.

Findings

Identifies parameter conditions for the Dagum function to be positive definite.

Establishes relationships with other families of monotonic functions.

Provides a comprehensive analysis of the Dagum correlation functions.

Abstract

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.