On two extensions of the canonical Feller–Spitzer distribution

Journal of Statistical Distributions and Applications

Vladimir Vladimirovich Vinogradov¹  Richard Bruce Paris² 

[1] Department of Mathematics, Ohio University, Athens, OH, USA;Division of Computing and Mathematics, Abertay University, Dundee, UK;

关键词: Bessel functions;    Continuous–time Bernoulli random walk;    Exponential convergence;    Feller–Spitzer distribution;    Hypergeometric functions;    Kurtosis;    Laplace transform;    Letac–Mora reciprocity;    Lévy measure;    Monotone density;    Natural exponential family;    Power tail;    Skewness;    Stochastically monotone family;    Variance function;    60E07 60E10 60F05;   

DOI  :  10.1186/s40488-021-00113-4

来源: Springer

【 摘 要 】

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.

【 授权许可】

CC BY