In probability and statistics, the generalized beta distribution[1] 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:
\( GB(y;a,b,c,p,q)={\frac {|a|y^{ap-1}(1-(1-c)(y/b)^{a})^{q-1}}{b^{ap}B(p,q)(1+c(y/b)^{a})^{p+q}}}\quad \quad {\text{ for }}0<y^{a}<{\frac {b^{a}}{1-c}}, \)
and zero otherwise. Here the parameters satisfy \( 0\leq c\leq 1 \) and b, p, and q positive. The function B(p,q) is the beta function.
GB distribution tree
Properties
Moments
It can be shown that the hth moment can be expressed as follows:
\( \operatorname {E} _{GB}(Y^{h})={\frac {b^{h}B(p+h/a,q)}{B(p,q)}}{}_{2}F_{1}{\begin{bmatrix}p+h/a,h/a;c\\p+q+h/a;\end{bmatrix}}, \)
where \( {}_{2}F_{1} \) 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 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:
\( GB1(y;a,b,p,q)={\frac {|a|y^{ap-1}(1-(y/b)^{a})^{q-1}}{b^{ap}B(p,q)}} \)
for \( <b^{a}} 0<y^{a}<b^{a} \) 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
\( \operatorname {E} _{GB1}(Y^{h})={\frac {b^{h}B(p+h/a,q)}{B(p,q)}}. \)
The GB1 includes the beta of the first kind (B1), generalized gamma(GG), and Pareto as special cases:
\( B1(y;b,p,q)=GB1(y;a=1,b,p,q), \)
\( GG(y;a,\beta ,p)=\lim _{q\to \infty }GB1(y;a,b=q^{1/a}\beta ,p,q), \)
\( PARETO(y;b,p)=GB1(y;a=-1,b,p,q=1). \)
Generalized beta of the second kind (GB2)
The GB2 is defined by the following pdf:
\( GB2(y;a,b,p,q)={\frac {|a|y^{ap-1}}{b^{ap}B(p,q)(1+(y/b)^{a})^{p+q}}} \)
for \( 0<y<\infty \) 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
\( \operatorname {E} _{GB2}(Y^{h})={\frac {b^{h}B(p+h/a,q-h/a)}{B(p,q)}}. \)
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 distribution (B) is defined by:[1]
\( B(y;b,c,p,q)={\frac {y^{p-1}(1-(1-c)(y/b))^{q-1}}{b^{p}B(p,q)(1+c(y/b))^{p+q}}} \)
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 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.
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 thehth moments given by
\( \operatorname {E} (Y_{GG}^{h})={\frac {\beta ^{h}\Gamma (p+h/a)}{\Gamma (p)}}. \)
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:
\( {\displaystyle PA(y;\beta ,\theta )=\lim _{a\rightarrow -\infty }GG(y;a,\beta ,p=-\theta /a)=\lim _{a\rightarrow -\infty }\left({\frac {\theta y^{-\theta -1}e^{-(y/\beta )^{a}}}{\beta ^{-\theta }(-\theta /a)\Gamma (-\theta /a)}}\right)=} \)
\( {\displaystyle \lim _{a\rightarrow -\infty }\left({\frac {\theta y^{-\theta -1}e^{-(y/\beta )^{a}}}{\beta ^{-\theta }\Gamma (1-\theta /a)}}\right)={\frac {\theta y^{-\theta -1}}{\beta ^{-\theta }}}} for β < y {\displaystyle \beta <y} {\displaystyle \beta <y} and 0 {\displaystyle 0} {\displaystyle 0} otherwise. \)
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 \( \theta >0. \)
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 hth moments are given by
\( {\displaystyle \operatorname {E} (Y_{ALL}^{h})={\frac {b^{h}\lambda _{1}\lambda _{2}}{(\lambda _{1}+h)(\lambda _{2}-h)}}.} \)
When \( \lambda_1 = \lambda_2, \) this is equivalent to the log-Laplace distribution.
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:
\( EGB(z;\delta ,\sigma ,c,p,q)={\frac {e^{p(z-\delta )/\sigma }(1-(1-c)e^{(z-\delta )/\sigma })^{q-1}}{|\sigma |B(p,q)(1+ce^{(z-\delta )/\sigma })^{p+q}}} \)
for \( -\infty <{\frac {z-\delta }{\sigma }}<\ln({\frac {1}{1-c}}) \), 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.[11]
The EGB family of distributions
Moment generating function
Using similar notation as above, the moment-generating function of the EGB can be expressed as follows:
\( M_{EGB}(Z)={\frac {e^{\delta t}B(p+t\sigma ,q)}{B(p,q)}}{}_{2}F_{1}{\begin{bmatrix}p+t\sigma ,t\sigma ;c\\p+q+t\sigma ;\end{bmatrix}}. \)
Multivariate generalized beta distribution
A multivariate generalized beta pdf extends the univariate distributions listed above. For 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 \( b_{i} \) and \( p_{i} \) is positive, and \( {\displaystyle 0} \) \( \leq \) \( c_{i} \) \( \leq \)( 1. The parameter q is assumed to be positive, and define the function \( {\displaystyle B(p_{1},...,p_{n},q)} \) = \( {\displaystyle {\frac {\Gamma (p_{1})...\Gamma (p_{n})\Gamma (q)}{\Gamma ({\bar {p}}+q)}}} \) for \( \bar{p} \)= \( {\displaystyle \sum _{i=1}^{n}p_{i}}. \)
The pdf of the multivariate generalized beta ( \( {\displaystyle MGB} \) ) may be written as follows:
\( {\displaystyle MGB(y;a,b,p,q,c)={\frac {(\prod _{i=1}^{n}|a_{i}|y_{i}^{a_{i}p_{i}-1})(1-\sum _{i=1}^{n}(1-c_{i})({\frac {y_{i}}{b_{i}}})^{a_{i}})^{q-1}}{(\prod _{i=1}^{n}b_{i}^{a_{i}p_{i}})B(p_{1},...,p_{n},q)(1+\sum _{i=1}^{n}c_{i}({\frac {y_{i}}{b_{i}}})^{a_{i}})^{{\bar {p}}+q}}}} \)
where \( {\displaystyle 0} \) < \( {\displaystyle \sum _{i=1}^{n}(1-c_{i})({\frac {y_{i}}{b_{i}}})^{a_{i}}} \) < 1 for \( {\displaystyle 0} \) \( \leq \) \( c_{i} \) < 1 and < \( y_{i} \) when \( c_{i} \) = 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]
Multivariate generalized beta of the first kind (MGB1)
When each \( c_{i} \) 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 \( {\displaystyle 0} \) < \( {\displaystyle \sum _{i=1}^{n}({\frac {y_{i}}{b_{i}}})^{a_{i}}} \) < .
Multivariate generalized beta of the second kind (MGB2)
In the case where each \( c_{i} \) 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 \( {\displaystyle 0}\) < \( y_{i} \) for all \( y_{i}. \)
Multivariate generalized gamma
The multivariate generalized gamma (MGG) pdf can be derived from the MGB pdf by substituting \( b_{i} \) = \( {\displaystyle \beta _{i}q^{\frac {1}{a_{i}}}} \) and taking the limit as q \( \to \) \( \infty \), 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 M \( {\displaystyle MGB2(y;a=1,b=1,p,q)} \) , 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:
\( {\displaystyle GB1(y_{i};a_{i},b_{i},p_{i},{\bar {p}}-p_{i}+q)={\frac {|a_{i}|y_{i}^{a_{i}p_{i}-1}(1-({\frac {y_{i}}{b_{i}}})^{a_{i}})^{{\bar {p}}-p_{i}+q-1}}{b_{i}^{a_{i}p_{i}}B(p_{i},{\bar {p}}-p_{i}+q)}}} \)
\( {\displaystyle GB2(y_{i};a_{i},b_{i},p_{i},q)={\frac {|a_{i}|y_{i}^{a_{i}p_{i}-1}}{b_{i}^{a_{i}p_{i}}B(p_{i},q)(1+({\frac {y_{i}}{b_{i}}})^{a_{i}})^{p_{i}+q}}}} \)
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]
\( {\begin{aligned}G=\left({\frac {1}{2\mu }}\right)\operatorname {E} (|Y-X|)=\left(P{\frac {1}{2\mu }}\right)\int _{0}^{\infty }\int _{0}^{\infty }|x-y|f(x)f(y)\,dxdy\\=1-{\frac {\int _{0}^{\infty }(1-F(y))^{2}\,dy}{\int _{0}^{\infty }(1-F(y))\,dy}}\\P=\left({\frac {1}{2\mu }}\right)\operatorname {E} (|Y-\mu |)=\left({\frac {1}{2\mu }}\right)\int _{0}^{\infty }|y-\mu |f(y)\,dy\\T=\operatorname {E} (\ln(Y/\mu )^{Y/\mu })=\int _{0}^{\infty }(y/\mu )\ln(y/\mu )f(y)\,dy\end{aligned}} \)
Hazard Functions
The hazard function, h(s), where f(s) is a pdf and F(s) the corresponding cdf, is defined by
\( h(s)={\frac {f(s)}{1-F(s)}} \)
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
References
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
Patil, G.P., Boswell, M.T., and Ratnaparkhi, M.V., Dictionary and Classified Bibliography of Statistical Distributions in Scientific Work Series, editor G.P. Patil, Internal Co-operative Publishing House, Burtonsville, Maryland, 1984.
Venter, G., Transformed beta and gamma distributions and aggregate losses, Proceedings of the Casualty Actuarial Society, 1983.
Kalbfleisch, J.D. and R.L. Prentice, The Statistical Analysis of Failure Time Data, New York: J. Wiley, 1980
Arnold, B.C., Pareto Distributions, Volume 5 in Statistical Distributions in Scientific Work Series, International Co-operative Publishing House, Burtonsville, Md. 1983.
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.
Stacy, E.W. (1962). "A Generalization of the Gamma Distribution." The Annals of Mathematical Statistics 33(3): 1187-1192. JSTOR 2237889
Reed, W.J. (2001). "The Pareto, Zipf, and other power laws." Economics Letters 74: 15-19. doi:10.1016/S0165-1765(01)00524-9
Higbee, J.D., Jensen, J.E., and McDonald, J.B. (2019). "The asymmetric log-Laplace distribution as a limiting case of the generalized beta distribution."Statistics and Probability Letters 151: 73-78. doi:10.1016/j.spl.2019.03.018
McDonald, James B. & Kerman, Sean C. (2013) "Skewness-Kurtosis Bounds for EGB1, EGB2, and Special Cases," Forthcoming
William M. Cockriel & James B. McDonald (2017): Two multivariate generalized beta families, Communications in Statistics - Theory and Methods, doi:10.1080/03610926.2017.1400058
Thurow, L.C. (1970) "Analyzing the American Income Distribution," Papers and Proceedings, American Economics Association, 60, 261-269
Dagum, C. (1977) "A New Model for Personal Income Distribution: Specification and Estimation," Economie Applique'e, 30, 413-437
Singh, S.K. and Maddala, G.S (1976) "A Function for the Size Distribution of Incomes," Econometrica, 44, 963-970
McDonald, J.B. and Ransom, M. (2008) "The Generalized Beta Distribution as a Model for the Distribution of Income: Estimation of Related Measures of Inequality", Modeling the Distributions and Lorenz Curves, "Economic Studies in Inequality: Social Exclusion and Well-Being", Springer: New York editor Jacques Silber, 5, 147-166
Glaser, Ronald E. (1980) "Bathtub and Related Failure Rate Characterizations," Journal of the American Statistical Association, 75(371), 667-672 doi:10.1080/01621459.1980.10477530
McDonald, James B. (1987) "A general methodology for determining distributional forms with applications in reliability," Journal of Statistical Planning and Inference, 16, 365-376 doi:10.1016/0378-3758(87)90089-9
McDonald, J.B. and Richards, D.O. (1987) "Hazard Functions and Generalized Beta Distributions", IEEE Transactions on Reliability, 36, 463-466
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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Probability distributions (List)
Discrete univariate
with finite support
Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher soliton discrete uniform Zipf Zipf–Mandelbrot
Discrete univariate
with infinite support
beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Flory–Schulz Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta
Continuous univariate
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arcsine ARGUS Balding–Nichols Bates beta beta rectangular continuous Bernoulli Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate
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Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Fréchet gamma gamma/Gompertz generalized gamma generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared
scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared noncentral F Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull
discrete Weibull Wilks's lambda
Continuous univariate
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Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's SU Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate
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Multivariate (joint)
Discrete
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Continuous
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Matrix-valued
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Wishart
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univariate von Mises
wrapped normal
wrapped Cauchy
wrapped exponential
wrapped asymmetric Laplace
wrapped Lévy
Bivariate (spherical)
Kent
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bivariate von Mises
Multivariate
von Mises–Fisher
Bingham
Degenerate and singular
Degenerate
Dirac delta function
Singular
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Families
Circular compound Poisson elliptical exponential natural exponential location–scale maximum entropy mixture Pearson Tweedie wrapped
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