On Mixtures of Gamma Distributions, Distributions with Hyperbolically Monotone Densities and Generalized Gamma Convolutions (GGC)

• Autorzy: Sjödin Tord

Identyfikatory

DOI

10.37190/0208-4147.41.1.1

• Języki publikacji: EN

Abstrakty

EN

Let Y be a standard Gamma(k) distributed random variable (rv), k > 0, and let X be an independent positive rv. If X has a hyperbolically monotone density of order k (HM_k), then Y · X and Y/X are generalized gamma convolutions (GGC). This extends work by Roynette et al. and Behme and Bondesson. The same conclusion holds with Y replaced by a finite sum of independent gamma variables with sum of shape parameters at most k. Both results are applied to subclasses of GGC.

Słowa kluczowe

EN

gamma distribution; hyperbolically monotone function; Laplace transform; generalized gamma convolution; GGC

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2021
• Tom: Vol. 41, Fasc. 1
• Strony: 1--7
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Sjödin Tord

• Department of Mathematics and Mathematical Statistics, Umeå University, 90187 Umeå, Sweden

Bibliografia

• [1] A. Behme and L. Bondesson, A class of scale mixtures of gamma(k)-distributions that are generalized gamma convolutions, Bernoulli 23 (2017), 773-787.
• [2] L. Bondesson, Generalized Gamma Convolutions and Related Classes of Distributions and Densities, Lecture Notes in Statist. 76, Springer, New York, 1992.
• [3] L. Bondesson, A class of probability distributions that is closed with respect to addition as well as multiplication of independent random variables, J. Theoret. Probab. 28 (2015), 1063-1081.
• [4] W. Feller, An Introduction to Probability Theory and its Applications, Vol. II, Wiley, New York, 1966.
• [5] B. Roynette, P. Vallois, and M. Yor, A family of generalized gamma convoluted variables, Probab. Math. Statist. 29 (2009), 181-204.
• [6] R. L. Schilling, R. Song, and Z. Vondraček, Bernstein Functions, de Gruyter Stud. Math. 37, de Gruyter, Berlin, 2010.
• [7] F. W. Steutel and K. van Harn, Infinite Divisibility of Probability Distributions on the Real Line, Dekker, New York, 2004.
• [8] O. Thorin, On the infinite divisibility of the lognormal distribution, Scand. Actuarial J. 1977, 121-148.
• [9] O. Thorin, An extension of the notion of a generalized Γ-convolution, Scand. Actuarial J. 1978, 141-149.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).