Moments

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

${\displaystyle \operatorname {E} _{GB}(Y^{h})={\frac {b^{h}B(p+h/a,q)}{B(p,q)}}{}_{2}F_{1}{\begin{bmatrix}p+h/a,h/a;c\\p+q+h/a;\end{bmatrix}},}$

where ${\displaystyle {}_{2}F_{1}}$ denotes the hypergeometric series (which converges for all h if c<1, or for all h/a<q if c=1 ).

Generalized beta of first kind (GB1)

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 ${\displaystyle 0<y^{a}<b^{a}}$ 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

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

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

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

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

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

Generalized beta of the second kind (GB2)

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 ${\displaystyle 0<y<\infty }$ 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

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

The GB2 is also known as the Generalized 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 the Feller-Pareto (Arnold, 1983)^[5] distribution. The GB2 nests common distributions such as the generalized gamma (GG), Burr type 3, Burr type 12, Dagum, lognormal, Weibull, gamma, Lomax, F statistic, Fisk or Rayleigh, chi-square, half-normal, half-Student's t, exponential, asymmetric log-Laplace, log-Laplace, power function, and the log-logistic.^[6]

Beta

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 ${\displaystyle 0<y<b/(1-c)}$ 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 the Beta prime), which correspond to c = 0 and c = 1, respectively. Setting ${\displaystyle c=0}$, ${\displaystyle b=1}$ yields the standard two-parameter beta distribution.

Generalized Gamma

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

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

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

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

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

Pareto

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

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

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

Power

The power (P) distribution is the following limiting case of the generalized gamma:

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

${\displaystyle \lim _{a\rightarrow \infty }{\frac {\theta y^{\theta -1}}{\Gamma (p+1)\beta ^{\theta }}}e^{-(y/\beta )^{a}}=\lim _{a\rightarrow \infty }{\frac {\theta y^{\theta -1}}{\Gamma ({\frac {\theta }{a}}+1)\beta ^{\theta }}}e^{-(y/\beta )^{a}}={\frac {\theta y^{\theta -1}}{\beta ^{\theta }}},}$

which is equivalent to the power function distribution for ${\displaystyle 0\leq y\leq \beta }$ and ${\displaystyle \theta >0}$.

Asymmetric Log-Laplace

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

${\displaystyle ALL(y;b,\lambda _{1},\lambda _{2})=\lim _{a\rightarrow \infty }GB2(y;a,b,p=\lambda _{1}/a,q=\lambda _{2}/a)={\frac {\lambda _{1}\lambda _{2}}{y(\lambda _{1}+\lambda _{2})}}{\begin{cases}({\frac {y}{b}})^{\lambda _{1}}&{\mbox{for }}0<y<b\\({\frac {b}{y}})^{\lambda _{2}}&{\mbox{for }}y\geq b\end{cases}}}$

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

${\displaystyle \operatorname {E} (Y_{ALL}^{h})={\frac {b^{h}\lambda _{1}\lambda _{2}}{(\lambda _{1}+h)(\lambda _{2}-h)}}.}$

When ${\displaystyle \lambda _{1}=\lambda _{2}}$, this is equivalent to the log-Laplace distribution.

Moment generating function

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

${\displaystyle M_{EGB}(Z)={\frac {e^{\delta t}B(p+t\sigma ,q)}{B(p,q)}}{}_{2}F_{1}{\begin{bmatrix}p+t\sigma ,t\sigma ;c\\p+q+t\sigma$ ;\end{bmatrix}}.}

Multivariate generalized beta of the first kind (MGB1)

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 ${\displaystyle 0}$ ${\displaystyle <}$ ${\displaystyle \sum _{i=1}^{n}({\frac {y_{i}}{b_{i}}})^{a_{i}}}$ ${\displaystyle <}$ ${\displaystyle 1}$.

Multivariate generalized beta of the second kind (MGB2)

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:

${\displaystyle MGB2(y;a,b,p,q)={\frac {(\prod _{i=1}^{n}|a_{i}|y_{i}^{a_{i}p_{i}-1})}{(\prod _{i=1}^{n}b_{i}^{a_{i}p_{i}})B(p_{1},...,p_{n},q)(1+\sum _{i=1}^{n}({\frac {y_{i}}{b_{i}}})^{a_{i}})^{{\bar {p}}+q}}}}$

when ${\displaystyle 0}$ ${\displaystyle <}$ ${\displaystyle y_{i}}$ for all ${\displaystyle y_{i}}$.

Multivariate generalized gamma

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 ${\displaystyle q}$ ${\displaystyle \to }$ ${\displaystyle \infty }$, with Stirling's approximation for the gamma function, yielding the following function:

${\displaystyle MGG(y;a,\beta ,p)=({\frac {(\prod _{i=1}^{n}|a_{i}|y_{i}^{a_{i}p_{i}-1})}{(\prod _{i=1}^{n}\beta _{i}^{a_{i}p_{i}})\Gamma (p_{i})}})e^{-\sum _{i=1}^{n}({\frac {y_{i}}{\beta _{i}}})^{a_{i}}}=\prod _{i=1}^{n}GG(y_{i};a_{i},\beta _{i},p_{i})}$

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

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

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:

${\displaystyle GB1(y_{i};a_{i},b_{i},p_{i},{\bar {p}}-p_{i}+q)={\frac {|a_{i}|y_{i}^{a_{i}p_{i}-1}(1-({\frac {y_{i}}{b_{i}}})^{a_{i}})^{{\bar {p}}-p_{i}+q-1}}{b_{i}^{a_{i}p_{i}}B(p_{i},{\bar {p}}-p_{i}+q)}}}$

${\displaystyle GB2(y_{i};a_{i},b_{i},p_{i},q)={\frac {|a_{i}|y_{i}^{a_{i}p_{i}-1}}{b_{i}^{a_{i}p_{i}}B(p_{i},q)(1+({\frac {y_{i}}{b_{i}}})^{a_{i}})^{p_{i}+q}}}}$

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]

${\displaystyle {\begin{aligned}G=\left({\frac {1}{2\mu }}\right)\operatorname {E} (|Y-X|)=\left(P{\frac {1}{2\mu }}\right)\int _{0}^{\infty }\int _{0}^{\infty }|x-y|f(x)f(y)\,dxdy\\=1-{\frac {\int _{0}^{\infty }(1-F(y))^{2}\,dy}{\int _{0}^{\infty }(1-F(y))\,dy}}\\P=\left({\frac {1}{2\mu }}\right)\operatorname {E} (|Y-\mu |)=\left({\frac {1}{2\mu }}\right)\int _{0}^{\infty }|y-\mu |f(y)\,dy\\T=\operatorname {E} (\ln(Y/\mu )^{Y/\mu })=\int _{0}^{\infty }(y/\mu )\ln(y/\mu )f(y)\,dy\end{aligned}}}$

Hazard Functions

The hazard 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 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