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Template:Primary sources 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.

Definition

A generalized beta random variable, Y, is defined by the following probability density function:

$GB(y;a,b,c,p,q)=\frac{\left|a\right|y^{ap-1}(1-(1-c)(y/b{)}^a{)}^{q-1}}{b^{ap}B(p,q)(1+c(y/b{)}^a{)}^{p+q}}\text{ for }0<y^a<\frac{b^a}{1-c},$

and zero otherwise. Here the parameters satisfy $0\le c\le 1$ and $b$, $p$, and $q$ positive. The function B(p,q) is the beta function.

File:GBtree.jpg

GB distribution tree

Properties

Moments

It can be shown that the hth moment can be expressed as follows:

${\mathrm{E}}_{GB}(Y^h)=\frac{b^{h}B(p+h/a,q)}{B(p,q)}{}_2{F}_1\left[\begin{array}{c}p+h/a,h/a;c\\ p+q+h/a;\end{array}\right],$

where ${}_2{F}_1$ 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 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:

$GB1(y;a,b,p,q)=\frac{\left|a\right|y^{ap-1}(1-(y/b{)}^a{)}^{q-1}}{b^{ap}B(p,q)}$

for $0<y^a<b^a$ where $b$, $p$, and $q$ are positive. It is easily verified that

$GB1(y;a,b,p,q)=GB(y;a,b,c=0,p,q).$

The moments of the GB1 are given by

${\mathrm{E}}_{GB1}(Y^h)=\frac{b^{h}B(p+h/a,q)}{B(p,q)}.$

The GB1 includes the beta of the first kind (B1), generalized gamma(GG), and Pareto as special cases:

$B1(y;b,p,q)=GB1(y;a=1,b,p,q),$

$GG(y;a,\beta ,p)=\underset{q\to \mathrm{\infty }}{\lim }GB1(y;a,b=q^{1/a}\beta ,p,q),$

$PARETO(y;b,p)=GB1(y;a=-1,b,p,q=1).$

Generalized beta of the second kind (GB2)

The GB2 (also known as the Generalized Beta Prime) is defined by the following pdf:

$GB2(y;a,b,p,q)=\frac{\left|a\right|y^{ap-1}}{b^{ap}B(p,q)(1+(y/b{)}^a{)}^{p+q}}$

for $0<y<\mathrm{\infty }$ and zero otherwise. One can verify that

$GB2(y;a,b,p,q)=GB(y;a,b,c=1,p,q).$

The moments of the GB2 are given by

${\mathrm{E}}_{GB2}(Y^h)=\frac{b^{h}B(p+h/a,q-h/a)}{B(p,q)}.$

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.

Beta

The beta distribution (B) is defined by:Template:Cn

$B(y;b,c,p,q)=\frac{y^{p-1}(1-(1-c)(y/b){)}^{q-1}}{b^{p}B(p,q)(1+c(y/b){)}^{p+q}}$

for $0<y<b/(1-c)$ and zero otherwise. Its relation to the GB is seen below:

$B(y;b,c,p,q)=GB(y;a=1,b,c,p,q).$

The beta family includes the betas of the first and second kind (B1 and B2, where the B2 is also referred to as the Beta prime), which correspond to c = 0 and c = 1, respectively.

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 $Y\sim GB(y;a,b,c,p,q)$, the random variable $Z=\ln (Y)$, with re-parametrization, is distributed as an exponential generalized beta (EGB), with the following pdf:

$EGB(z;\delta ,\sigma ,c,p,q)=\frac{e^{p(z-\delta )/\sigma }(1-(1-c)e^{(z-\delta )/\sigma }{)}^{q-1}}{\left|\sigma \right|B(p,q)(1+c{e}^{(z-\delta )/\sigma }{)}^{p+q}}$

for $-\mathrm{\infty }<\frac{z-\delta }{\sigma }<\ln (\frac{1}{1-c})$, and zero otherwise. The EGB includes generalizations of the Gompertz, Gumbell, extreme value type I, logistic, Burr-2, exponential, and normal distributions.

Included is a figure showing the relationship between the EGB and its special and limiting cases (see McDonald and Xu (1995) ).

File:EGBtree.jpg

EGB distribution tree

Moment generating function

Using similar notation as above, the moment-generating function of the EGB can be expressed as follows:

$M_{EGB}(Z)=\frac{e^{\delta t}B(p+t\sigma ,q)}{B(p,q)}{}_2{F}_1\left[\begin{array}{c}p+t\sigma ,t\sigma ;c\\ p+q+t\sigma ;\end{array}\right].$

Uses

The flexibility provided by the GB family is used in modeling the distribution of:Template:Cn

• family income
• stock returns
• insurance losses

Applications involving members of the EGB family include:Template:Cn

• partially adaptive estimation of regression
• time series models

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.

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