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Inprobability andstatistics, thegeneralized beta distribution^[1] is acontinuous probability distribution with fourshape parameters, including more than thirty named distributions aslimiting orspecial cases. A fifth parameter forscaling is sometimes included, while a sixth parameter forlocation is customarily left implicit and excluded from the characterization. The distribution has been used in the modeling ofincome distribution, stock returns, as well as inregression analysis. Theexponential 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 (pdf):

and zero otherwise. Here the parameters satisfy${\displaystyle a\ne 0}$,${\displaystyle 0\le c\le 1}$ and${\displaystyle b}$,${\displaystyle p}$, and${\displaystyle q}$ positive. The functionB(p,q) is thebeta function. The parameter${\displaystyle b}$ is thescale parameter and can thus be set to${\displaystyle 1}$without loss of generality, but it is usually made explicit as in the function above. Thelocation parameter (not included in the formula above) is usually left implicit and set to${\displaystyle 0}$.

It can be shown that thehth moment can be expressed as follows:

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

where${\displaystyle {}_2{F}_1}$ denotes thehypergeometric series (which converges for allh ifc < 1, or for allh /a <q ifc = 1 ).

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.

The generalized beta of the first kind is defined by the following pdf:

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

for where${\displaystyle b}$,${\displaystyle p}$, and${\displaystyle q}$ are positive. It is easily verified that

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

The moments of the GB1 are given by

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

The GB1 includes thebeta of the first kind (B1),generalized gamma (GG), andPareto (PA) as special cases:

${\displaystyle B1(y;b,p,q)=GB1(y;a=1,b,p,q),}$

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

${\displaystyle PA(y;b,p)=GB1(y;a=-1,b,p,q=1).}$

The GB2 is defined by the following pdf:

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

for and zero otherwise. One can verify that

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

The moments of the GB2 are given by

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

The GB2 is also known as theGeneralized Beta Prime (Patil, Boswell, Ratnaparkhi (1984)),^[2] the transformed beta (Venter, 1983),^[3] the generalized F (Kalfleisch and Prentice, 1980),^[4] and is a special case (μ≡0) of theFeller-Pareto (Arnold, 1983)^[5] distribution. The GB2 nests common distributions such as thegeneralized gamma (GG), Burr type 3,Burr type 12,Dagum,lognormal,Weibull,gamma,Lomax,F statistic, Fisk orRayleigh,chi-square,half-normal,half-Student's t,exponential, asymmetric log-Laplace,log-Laplace,power function, and thelog-logistic.^[6]

The beta family of distributions (B) is defined by:^[1]

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

for and zero otherwise. Its relation to the GB is seen below:

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

The beta family includes the beta of the first and second kind^[7] (B1 and B2, where the B2 is also referred to as theBeta prime), which correspond toc = 0 andc = 1, respectively. Setting${\displaystyle c=0}$,${\displaystyle b=1}$ yields the standard two-parameterbeta distribution.

Thegeneralized gamma distribution (GG) is a limiting case of the GB2. Its PDF is defined by:^[8]

${\displaystyle GG(y;a,\beta ,p)=\underset{q\to \mathrm{\infty }}{lim}GB2(y,a,b=q^{1/a}\beta ,p,q)=\frac{|a|y^{ap-1}{e}^{-(y/\beta {)}^a}}{{\beta }^{ap}\mathrm{\Gamma }(p)}}$

with the${\displaystyle h}$th moments given by

${\displaystyle \mathrm{E}(Y_{GG}^h)=\frac{{\beta }^h\mathrm{\Gamma }(p+h/a)}{\mathrm{\Gamma }(p)}.}$

As noted earlier, the GB distribution family tree visually depicts the special and limiting cases (see McDonald and Xu (1995) ).

ThePareto distribution (PA) is the following limiting case of the generalized gamma:

${\displaystyle PA(y;\beta ,\theta )=\underset{a\to -\mathrm{\infty }}{lim}GG(y;a,\beta ,p=-\theta /a)=\underset{a\to -\mathrm{\infty }}{lim}\left(\frac{\theta y^{-\theta -1}{e}^{-(y/\beta {)}^a}}{{\beta }^{-\theta }(-\theta /a)\mathrm{\Gamma }(-\theta /a)}\right)=}$

${\displaystyle \underset{a\to -\mathrm{\infty }}{lim}\left(\frac{\theta y^{-\theta -1}{e}^{-(y/\beta {)}^a}}{{\beta }^{-\theta }\mathrm{\Gamma }(1-\theta /a)}\right)=\frac{\theta y^{-\theta -1}}{{\beta }^{-\theta }}}$ for and${\displaystyle 0}$ otherwise.

Thepower function distribution (P) is the following limiting case of the generalized gamma:

${\displaystyle P(y;\beta ,\theta )=\underset{a\to \mathrm{\infty }}{lim}GG(y;a=\theta /p,\beta ,p)=\underset{a\to \mathrm{\infty }}{lim}\frac{\mid \frac{\theta }{p}|y^{\theta -1}{e}^{-(y/\beta {)}^a}}{{\beta }^{\theta }\mathrm{\Gamma }(p)}=\underset{a\to \mathrm{\infty }}{lim}\frac{\theta y^{\theta -1}}{p\mathrm{\Gamma }(p){\beta }^{\theta }}{e}^{-(y/\beta {)}^a}=}$

${\displaystyle \underset{a\to \mathrm{\infty }}{lim}\frac{\theta y^{\theta -1}}{\mathrm{\Gamma }(p+1){\beta }^{\theta }}{e}^{-(y/\beta {)}^a}=\underset{a\to \mathrm{\infty }}{lim}\frac{\theta y^{\theta -1}}{\mathrm{\Gamma }(\frac{\theta }{a}+1){\beta }^{\theta }}{e}^{-(y/\beta {)}^a}=\frac{\theta y^{\theta -1}}{{\beta }^{\theta }}}$ for and${\displaystyle \theta >0}$.

The asymmetric log-Laplace distribution (also referred to as the double Pareto distribution^[9]) is defined by:^[10]

where the${\displaystyle h}$th moments are given by

${\displaystyle \mathrm{E}(Y_{ALL}^h)=\frac{b^h{\lambda }_1{\lambda }_2}{({\lambda }_1+h)({\lambda }_2-h)}.}$

When${\displaystyle {\lambda }_1={\lambda }_2}$, this is equivalent to thelog-Laplace distribution.

Letting${\displaystyle Y\sim GB(y;a,b,c,p,q)}$ (without location parameter), the random variable${\displaystyle Z=\ln (Y)}$, with re-parametrization${\displaystyle \delta =\ln (b)}$ and${\displaystyle \sigma =1/a}$, is distributed as an exponential generalized beta (EGB), with the following pdf:

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

for, and zero otherwise.The EGB includes generalizations of theGompertz,Gumbel,extreme value type I,logistic, Burr-2,exponential, andnormal distributions. The parameter${\displaystyle \delta =\ln (b)}$ is thelocation parameter of the EGB (while${\displaystyle b}$ is thescale parameter of the GB), and${\displaystyle \sigma =1/a}$ is thescale parameter of the EGB (while${\displaystyle a}$ is ashape parameter of the GB); The EGB has thus threeshape parameters.

Included is a figure showing the relationship between the EGB and its special and limiting cases.^[11]

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

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

A multivariate generalized beta pdf extends the univariate distributions listed above. For${\displaystyle n}$ variables${\displaystyle y=(y_1,...,y_n)}$, define${\displaystyle 1xn}$ parameter vectors by${\displaystyle a=(a_1,...,a_n)}$,${\displaystyle b=(b_1,...,b_n)}$,${\displaystyle c=(c_1,...,c_n)}$, and${\displaystyle p=(p_1,...,p_n)}$ where each${\displaystyle b_i}$ and${\displaystyle p_i}$ is positive, and${\displaystyle 0}$${\displaystyle \le }$${\displaystyle c_i}$${\displaystyle \le }$${\displaystyle 1}$. The parameter${\displaystyle q}$ is assumed to be positive, and define the function${\displaystyle B(p_1,...,p_n,q)}$ =${\displaystyle \frac{\mathrm{\Gamma }(p_1)...\mathrm{\Gamma }(p_n)\mathrm{\Gamma }(q)}{\mathrm{\Gamma }(\overline{p}+q)}}$ for${\displaystyle \overline{p}}$ =${\displaystyle \sum _{i=1}^n{p}_i}$.

The pdf of the multivariate generalized beta (${\displaystyle MGB}$) may be written as follows:

${\displaystyle MGB(y;a,b,p,q,c)=\frac{(\prod _{i=1}^n|a_i|y_i^{a_i{p}_i-1})(1-\sum _{i=1}^n(1-c_i)(\frac{y_i}{b_i}{)}^{a_i}{)}^{q-1}}{(\prod _{i=1}^n{b}_i^{a_i{p}_i})B(p_1,...,p_n,q)(1+\sum _{i=1}^n{c}_i(\frac{y_i}{b_i}{)}^{a_i}{)}^{\overline{p}+q}}}$

where${\displaystyle 0}$${\displaystyle \sum _{i=1}^n(1-c_i)(\frac{y_i}{b_i}{)}^{a_i}}$${\displaystyle 1}$ for${\displaystyle 0}$${\displaystyle \le }$${\displaystyle c_i}$${\displaystyle 1}$ and${\displaystyle 0}$${\displaystyle y_i}$ when${\displaystyle c_i}$ =${\displaystyle 1}$.

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]

When each${\displaystyle c_i}$ is equal to 0, the MGB function simplifies to the multivariate generalized beta of the first kind (MGB1), which is defined by:

${\displaystyle MGB1(y;a,b,p,q)=\frac{(\prod _{i=1}^n|a_i|y_i^{a_i{p}_i-1})(1-\sum _{i=1}^n(\frac{y_i}{b_i}{)}^{a_i}{)}^{q-1}}{(\prod _{i=1}^n{b}_i^{a_i{p}_i})B(p_1,...,p_n,q)}}$

where${\displaystyle 0}$${\displaystyle \sum _{i=1}^n(\frac{y_i}{b_i}{)}^{a_i}}$${\displaystyle 1}$.

In the case where each${\displaystyle c_i}$ is equal to 1, the MGB simplifies to the multivariate generalized beta of the second kind (MGB2), with the pdf defined below:

${\displaystyle MGB2(y;a,b,p,q)=\frac{(\prod _{i=1}^n|a_i|y_i^{a_i{p}_i-1})}{(\prod _{i=1}^n{b}_i^{a_i{p}_i})B(p_1,...,p_n,q)(1+\sum _{i=1}^n(\frac{y_i}{b_i}{)}^{a_i}{)}^{\overline{p}+q}}}$

when${\displaystyle 0}$${\displaystyle y_i}$ for all${\displaystyle y_i}$.

The multivariate generalized gamma (MGG) pdf can be derived from the MGB pdf by substituting${\displaystyle b_i}$ =${\displaystyle {\beta }_i{q}^{\frac{1}{a_i}}}$ and taking the limit as${\displaystyle q}$${\displaystyle \to }$${\displaystyle \mathrm{\infty }}$, with Stirling's approximation for the gamma function, yielding the following function:

${\displaystyle MGG(y;a,\beta ,p)=(\frac{(\prod _{i=1}^n|a_i|y_i^{a_i{p}_i-1})}{(\prod _{i=1}^n{\beta }_i^{a_i{p}_i})\mathrm{\Gamma }(p_i)})e^{-\sum _{i=1}^n(\frac{y_i}{{\beta }_i}{)}^{a_i}}=\prod _{i=1}^{n}GG(y_i;a_i,{\beta }_i,p_i)}$

which is the product of independently but not necessarily identically distributed generalized gamma random variables.

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 theDirichlet distribution (standard form) given by${\displaystyle MGB1(y;a=1,b=1,p,q)}$, themultivariate inverted beta andinverted Dirichlet (Dirichlet type 2) distribution given by${\displaystyle MGB2(y;a=1,b=1,p,q)}$, and the multivariate Burr distribution given by${\displaystyle MGB2(y;a,b,p,q=1)}$.

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:

${\displaystyle GB1(y_i;a_i,b_i,p_i,\overline{p}-p_i+q)=\frac{|a_i|y_i^{a_i{p}_i-1}(1-(\frac{y_i}{b_i}{)}^{a_i}{)}^{\overline{p}-p_i+q-1}}{b_i^{a_i{p}_i}B(p_i,\overline{p}-p_i+q)}}$

${\displaystyle GB2(y_i;a_i,b_i,p_i,q)=\frac{|a_i|y_i^{a_i{p}_i-1}}{b_i^{a_i{p}_i}B(p_i,q)(1+(\frac{y_i}{b_i}{)}^{a_i}{)}^{p_i+q}}}$

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

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 theGini index (G), Pietra index (P), andTheil index (T) can be expressed in terms of the distributional parameters, as given by McDonald and Ransom (2008):^[16]

${\displaystyle \begin{array}{r}G=\left(\frac{1}{2\mu }\right)\mathrm{E}(|Y-X|)=\left(P\frac{1}{2\mu }\right){\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}|x-y|f(x)f(y)dxdy\\ =1-\frac{{\int }_0^{\mathrm{\infty }}(1-F(y){)}^{2}dy}{{\int }_0^{\mathrm{\infty }}(1-F(y))dy}\\ P=\left(\frac{1}{2\mu }\right)\mathrm{E}(|Y-\mu |)=\left(\frac{1}{2\mu }\right){\int }_0^{\mathrm{\infty }}|y-\mu |f(y)dy\\ T=\mathrm{E}(\ln (Y/\mu {)}^{Y/\mu })={\int }_0^{\mathrm{\infty }}(y/\mu )\ln (y/\mu )f(y)dy\end{array}}$

Thehazard function, h(s), where f(s) is a pdf and F(s) the corresponding cdf, is defined by

${\displaystyle h(s)=\frac{f(s)}{1-F(s)}}$

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 thegeneralized 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. Thegeneralized 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]

• 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.

• Continuous distributions

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