< List of probability distributions < Kummer distribution
The Kummer distribution was derived by Armero and Bayarri [1, 2], who derived it as a posterior distribution of certain basic parameters in a Bayesian analysis of a queuing system. Today, it is a major player in probability theory, providing solutions to some characterization problems and investigation of the Matsumoto-Yor type independence property.
It is named after the inclusion of one of two Kummer’s functions U, aka confluent hypergeometric functions of the second kind, in its density function. This is a standard function in many popular mathematical software packages.
Properties of the Kummer distribution
A continuous random variable has a Kummer distribution if its probability density function (pdf) has the form [3]
where C, the proportionality constant, is Γ(α)/δα U(α, α + 1 – γ, Β/δ)
- When γ = 0, the distribution becomes the gamma distribution with shape parameter α and scale parameter β [4].
- When δ = 1, the distribution can be viewed as scale-standard.
The Pareto distribution can be derived as a special case of the Kummer distribution.
References
- C. Armero and M. J. Bayarri, ‘A Bayesian analysis of a queuing system with unlimited service’, J. Statist. Plann. Inference 58 (1997), 241–264
- Konzu, E. (2021). Rate of convergence in total variation for the generalized inverse Gaussian and the Kummer distributions. Open Journal of Mathematical Sciences. Vol. 5 (2021), Issue 1, pp. 182 – 191 DOI: 10.30538/oms2021.0155
- C. Armero and M. J. Bayarri, ‘A Bayesian analysis of a queuing system with unlimited service’, Technical Report No. 93–50, (Department of Statistics, Purdue University, 1993).
- Nagar, D. & Gupta, A. Matrix-Variate Kummer-Beta Distribution. Journal of the Australian Mathematics Society 73 (2002), 11-25.