期刊
BERNOULLI
卷 14, 期 4, 页码 1134-1149出版社
INT STATISTICAL INST
DOI: 10.3150/08-BEJ139
关键词
Bernstein function; completely monotonic function; Dagum family; isotropy; logarithmically completely monotonic function; Stieltjes transform
资金
- Spanish Ministry of Science and Education [MTM2007-62923]
A function rho: [0, infinity) -> (0, 1] is a completely monotonic function if and only if rho(||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 rho(beta, gamma)(x) = 1 - (x(beta)/1+x(beta))(gamma), x >= 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.
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