Generalized beta distribution
|This article relies on references to primary sources. (April 2013)|
In probability and statistics, the generalized beta distribution is a continuous probability distribution with five parameters, including more than thirty named distributions as limiting or special cases. It has been used in the modeling of income distribution, stock returns, as well as in regression analysis. The exponential generalized Beta (EGB) distribution follows directly from the GB and generalizes other common distributions.
A generalized beta random variable, Y, is defined by the following probability density function:
and zero otherwise. Here the parameters satisfy and , , and positive. The function B(p,q) is the beta function.
It can be shown that the hth moment can be expressed as follows:
where denotes the hypergeometric series (which converges for all h if c<1, or for all h/a<q if c=1 ).
The generalized beta encompasses a number of distributions in its family as special cases. Listed below are its three direct descendants, or sub-families.
Generalized beta of first kind (GB1)
The generalized beta of the first kind is defined by the following pdf:
for where , , and are positive. It is easily verified that
The moments of the GB1 are given by
Generalized beta of the second kind (GB2)
The GB2 (also known as the Generalized Beta Prime) is defined by the following pdf:
for and zero otherwise. One can verify that
The moments of the GB2 are given by
The GB2 nests common distributions such as the generalized gamma (GG), Burr type 3, Burr type 12, lognormal, Weibull, gamma, Lomax, F statistic, Fisk or Rayleigh, chi-square, half-normal, half-Student's, exponential, and the log-logistic.
for and zero otherwise. Its relation to the GB is seen below:
A figure showing the relationship between the GB and its special and limiting cases is included above (see McDonald and Xu (1995) ).
Exponential generalized beta distribution
Letting , the random variable , with re-parametrization, is distributed as an exponential generalized beta (EGB), with the following pdf:
Included is a figure showing the relationship between the EGB and its special and limiting cases (see McDonald and Xu (1995) ).
Moment generating function
Using similar notation as above, the moment-generating function of the EGB can be expressed as follows:
The flexibility provided by the GB family is used in modeling the distribution of:
- family income
- stock returns
- insurance losses
Applications involving members of the EGB family include:
- partially adaptive estimation of regression
- time series models
- McDonald, James B. & Xu, Yexiao J. (1995) "A generalization of the beta distribution with applications," Journal of Econometrics, 66(1–2), 133–152 doi:10.1016/0304-4076(94)01612-4
- McDonald, J.B. (1984) "Some generalized functions for the size distributions of income", Econometrica 52, 647–663.
- Stuart, A. and Ord, J.K. (1987): Kendall's Advanced Theory of Statistics, New York: Oxford University Press.
- C. Kleiber and S. Kotz (2003) Statistical Size Distributions in Economics and Actuarial Sciences. New York: Wiley
- Johnson, N. L., S. Kotz, and N. Balakrishnan (1994) Continuous Univariate Distributions. Vol. 2, Hoboken, NJ: Wiley-Interscience.