A generalized Planck distribution. (English) Zbl 1110.62015
Summary: We introduce a generalization of the standard Planck distribution discussed by N. L. Johnson and S. Kotz [Distributions in statistics: Continuous univariate distributions. Vol. 2, Section 33.6.1 (1970; Zbl 0213.21101)]. This generalization results in a very flexible family which contains the gamma distribution as a particular case. We provide a comprehensive treatment of the mathematical properties of the family. We derive expressions for the \(n\)th moment, moment generating function, characteristic function, mean deviation about the mean, mean deviation about the median, Rényi entropy, Shannon entropy and the asymptotic distribution of extreme order statistics. Estimation and simulation issues are also considered.
MSC:
| 62E10 | Characterization and structure theory of statistical distributions |
| 62E20 | Asymptotic distribution theory in statistics |
Keywords:
gamma distributionCitations:
Zbl 0213.21101References:
| [1] | Dudewicz, E. J. andMishra, S. N. (1988).Modern Mathematical Statistics, John Wiley & Sons, New York. |
| [2] | Gradshteyn, I. S. andRyzhik, I. M. (2000).Table of Integrals, Series, and Products. Academic Press. San Diego, 6th ed. · Zbl 0981.65001 |
| [3] | Johnson, N. L. andKotz, S. (1970).Distributions in Statistics: Continuous Univariate Distributions, Vol. 2, John Wiley & Sons, New York. |
| [4] | Leadbetter, M. R., Lindgren, G., andRootzén, H. (1987).Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York. |
| [5] | Rényi, A. (1961). On measures of entropy and information. InProceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, pp. 547–561, University of California Press, Berkeley. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
