On two extensions of the canonical Feller–Spitzer distribution

• Published in: Journal of statistical distributions and applications Vol. 8; no. 1
• Main Authors: Vinogradov, Vladimir Vladimirovich, Paris, Richard Bruce
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 04-03-2021; Springer Nature B.V
• Subjects: Bessel functions; Humanities and Social Sciences; Hypergeometric functions; Kurtosis; Laplace transforms; Mathematics and Statistics; multidisciplinary; Parameters; Science; Skewness; Statistical Theory and Methods; Statistics; Statistics and Computing/Statistics Programs; Third International Conference on Statistical Distributions and Applications; Hypergeometric functions; Exponential convergence; Feller–Spitzer distribution; Letac–Mora reciprocity; Variance function; Stochastically monotone family; Power tail; Kurtosis; 60E07 60E10 60F05; Bessel functions; Natural exponential family; Monotone density; Continuous–time Bernoulli random walk; Lévy measure; Skewness; Laplace transform
• ISSN: 2195-5832; 2195-5832
• DOI: 10.1186/s40488-021-00113-4

We introduce two extensions of the canonical Feller–Spitzer distribution from the class of Bessel densities, which comprise two distinct stochastically decreasing one-parameter families of positive absolutely continuous infinitely divisible distributions with monotone densities, whose upper tails exhibit a power decay. The densities of the members of the first class are expressed in terms of the modified Bessel function of the first kind, whereas the members of the second class have the densities of their Lévy measure given by virtue of the same function. The Laplace transforms for both these families possess closed–form representations in terms of specific hypergeometric functions. We obtain the explicit expressions by virtue of the particular parameter value for the moments of the distributions considered and establish the monotonicity of the mean, variance, skewness and excess kurtosis within the families. We derive numerous properties of members of these classes by employing both new and previously known properties of the special functions involved and determine the variance function for the natural exponential family generated by a member of the second class.