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. The parameter is the scale parameter and can thus be set to without loss of generality, but it is usually made explicit as in the function above (while the location parameter is usually left implicit and set to as in the function above).
GB distribution tree
Properties
Moments
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 ).
Related distributions
The generalized beta encompasses many distributions as limiting or special cases. These are depicted in the GB distribution tree shown above. 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 beta family of distributions (B) is defined by:[1]
for and zero otherwise. Its relation to the GB is seen below:
The beta family includes the beta of the first and second kind[7] (B1 and B2, where the B2 is also referred to as the Beta prime), which correspond to c = 0 and c = 1, respectively. Setting , yields the standard two-parameter beta distribution.
Letting (without location parameter), the random variable , with re-parametrization and , is distributed as an exponential generalized beta (EGB), with the following pdf:
A multivariate generalized beta pdf extends the univariate distributions listed above. For variables , define parameter vectors by , , , and where each and is positive, and . The parameter is assumed to be positive, and define the function = for = .
The pdf of the multivariate generalized beta () may be written as follows:
where for and when = .
Like the univariate generalized beta distribution, the multivariate generalized beta includes several distributions in its family as special cases. By imposing certain constraints on the parameter vectors, the following distributions can be easily derived.[12]
Multivariate generalized beta of the first kind (MGB1)
When each is equal to 0, the MGB function simplifies to the multivariate generalized beta of the first kind (MGB1), which is defined by:
where .
Multivariate generalized beta of the second kind (MGB2)
In the case where each is equal to 1, the MGB simplifies to the multivariate generalized beta of the second kind (MGB2), with the pdf defined below:
when for all .
Multivariate generalized gamma
The multivariate generalized gamma (MGG) pdf can be derived from the MGB pdf by substituting = and taking the limit as , with Stirling's approximation for the gamma function, yielding the following function:
which is the product of independently but not necessarily identically distributed generalized gamma random variables.
Other multivariate distributions
Similar pdfs can be constructed for other variables in the family tree shown above, simply by placing an M in front of each pdf name and finding the appropriate limiting and special cases of the MGB as indicated by the constraints and limits of the univariate distribution. Additional multivariate pdfs in the literature include the Dirichlet distribution (standard form) given by , the multivariate inverted beta and inverted Dirichlet (Dirichlet type 2) distribution given by , and the multivariate Burr distribution given by .
Marginal density functions
The marginal density functions of the MGB1 and MGB2, respectively, are the generalized beta distributions of the first and second kind, and are given as follows:
Applications
The flexibility provided by the GB family is used in modeling the distribution of:
distribution of income
hazard functions
stock returns
insurance losses
Applications involving members of the EGB family include:[1][6]
partially adaptive estimation of regression models
time series models
(G)ARCH models
Distribution of Income
The GB2 and several of its special and limiting cases have been widely used as models for the distribution of income. For some early examples see Thurow (1970),[13] Dagum (1977),[14] Singh and Maddala (1976),[15] and McDonald (1984).[6]Maximum likelihood estimations using individual, grouped, or top-coded data are easily performed with these distributions.
Measures of inequality, such as the Gini index (G), Pietra index (P), and Theil index (T) can be expressed in terms of the distributional parameters, as given by McDonald and Ransom (2008):[16]
Hazard Functions
The hazard function, h(s), where f(s) is a pdf and F(s) the corresponding cdf, is defined by
Hazard functions are useful in many applications, such as modeling unemployment duration, the failure time of products or life expectancy. Taking a specific example, if s denotes the length of life, then h(s) is the rate of death at age s, given that an individual has lived up to age s. The shape of the hazard function for human mortality data might appear as follows: decreasing mortality in the first few months of life, then a period of relatively constant mortality and finally an increasing probability of death at older ages.
Special cases of the generalized beta distribution offer more flexibility in modeling the shape of the hazard function, which can call for "∪" or "∩" shapes or strictly increasing (denoted by I}) or decreasing (denoted by D) lines. The generalized gamma is "∪"-shaped for a>1 and p<1/a, "∩"-shaped for a<1 and p>1/a, I-shaped for a>1 and p>1/a and D-shaped for a<1 and p>1/a.[17] This is summarized in the figure below.[18][19]
Possible hazard function shapes using the generalized gamma
References
Bibliography
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.