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| In [[probability and statistics]], the '''generalized beta distribution'''<ref>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}}</ref> is a [[continuous probability distribution]] with five parameters, including more than thirty named distributions as [[Limiting case|limiting]] or [[special case]]s. 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.
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| == Definition ==
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| A generalized beta random variable, ''Y'', is defined by the following probability density function:
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| :<math> 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} , </math>
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| and zero otherwise. Here the parameters satisfy <math> 0 \le c \le 1 </math> and <math> b </math>, <math> p </math>, and <math> q </math> positive. The function ''B''(''p,q'') is the [[beta function]]. | |
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| [[File:GBtree.jpg|thumb|GB distribution tree]]
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| == Properties ==
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| === Moments ===
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| It can be shown that the ''h''th moment can be expressed as follows:
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| :<math> \operatorname{E}_{GB}(Y^{h})=\frac{b^{h}B(p+h/a,q)}{B(p,q)}{}_{2}F_{1} \begin{bmatrix}
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| p + h/a,h/a;c \\
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| p + q +h/a;
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| \end{bmatrix},
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| </math> | |
| where <math>{}_{2}F_{1}</math> denotes the [[hypergeometric series]] (which converges for all ''h'' if ''c''<1, or for all ''h''/''a''<''q'' if ''c''=1 ).
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| == Related distributions ==
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| The generalized beta encompasses a number of distributions in its family as special cases. Listed below are its three direct descendants, or sub-families.
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| === Generalized beta of first kind (GB1) ===
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| The generalized beta of the first kind is defined by the following pdf:
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| :<math> GB1(y;a,b,p,q) = \frac{|a|y^{ap-1}(1-(y/b)^{a})^{q-1}}{b^{ap}B(p,q)} </math>
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| for <math> 0< y^{a}<b^{a} </math> where <math> b </math>, <math> p </math>, and <math> q </math> are positive. It is easily verified that
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| :<math> GB1(y;a,b,p,q) = GB(y;a,b,c=0,p,q). </math>
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| The moments of the GB1 are given by
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| :<math> \operatorname{E}_{GB1}(Y^{h}) = \frac{b^{h}B(p+h/a,q)}{B(p,q)}. </math>
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| The GB1 includes the [[Beta distribution|beta of the first kind]] (B1), [[Generalized gamma distribution|generalized gamma]](GG), and [[Pareto distribution|Pareto]] as special cases:
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| :<math> B1(y;b,p,q) = GB1(y;a=1,b,p,q) ,</math>
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| :<math> GG(y;a,\beta,p) = \lim_{q \to \infty}
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| GB1(y;a,b=q^{1/a}\beta,p,q) ,</math>
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| :<math> PARETO(y;b,p) = GB1(y;a=-1,b,p,q=1) . </math>
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| === Generalized beta of the second kind (GB2) ===
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| The GB2 (also known as the [[Generalized_beta_prime_distribution#Generalization|Generalized Beta Prime]]) is defined by the following pdf:
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| :<math> GB2(y;a,b,p,q) = \frac{|a|y^{ap-1}}{b^{ap}B(p,q)(1+(y/b)^a)^{p+q}} </math>
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| for <math> 0< y < \infty </math> and zero otherwise. One can verify that
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| :<math> GB2(y;a,b,p,q) = GB(y;a,b,c=1,p,q). </math>
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| The moments of the GB2 are given by
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| :<math> \operatorname{E}_{GB2}(Y^h) = \frac{b^h B(p+h/a,q-h/a)}{B(p,q)}. </math>
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| The GB2 nests common distributions such as the generalized gamma (GG), Burr type 3, Burr type 12, [[lognormal]], [[Weibull distribution|Weibull]], [[Gamma distribution|gamma]], [[Lomax distribution|Lomax]], [[F-distribution|F statistic]], Fisk or [[Rayleigh distribution|Rayleigh]], [[Chi-squared distribution|chi-square]], [[Half-normal distribution|half-normal]], half-Student's, [[Exponential distribution|exponential]], and the [[Log-logistic distribution|log-logistic]].<ref>McDonald, J.B. (1984) "Some generalized functions for the size distributions of income", ''Econometrica'' 52, 647–663.</ref>
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| === Beta ===
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| The [[beta distribution]] (B) is defined by:{{cn|date=April 2013}}
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| :<math> 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}} </math>
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| for <math> 0<y<b/(1-c) </math> and zero otherwise. Its relation to the GB is seen below:
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| :<math> B(y;b,c,p,q) = GB(y;a=1,b,c,p,q). </math>
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| The beta family includes the betas of the first and second kind<ref>Stuart, A. and Ord, J.K. (1987): Kendall's Advanced Theory of Statistics, New York: Oxford University Press.</ref> (B1 and B2, where the B2 is also referred to as the [[Beta prime distribution|Beta prime]]), which correspond to ''c'' = 0 and ''c'' = 1, respectively.
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| A figure showing the relationship between the GB and its special and limiting cases is included above (see McDonald and Xu (1995) ).
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| == Exponential generalized beta distribution ==
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| Letting <math> Y \sim GB(y;a,b,c,p,q) </math>, the random variable <math> Z = \ln(Y) </math>, with re-parametrization, is distributed as an exponential generalized beta (EGB), with the following pdf:
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| :<math> 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}}</math>
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| for <math> -\infty < \frac{z-\delta}{\sigma}<\ln(\frac{1}{1-c}) </math>, and zero otherwise.
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| The EGB includes generalizations of the [[Gompertz distribution|Gompertz]], [[Gumbel distribution|Gumbell]], [[Type I extreme value distribution|extreme value type I]], [[Logistic distribution|logistic]], Burr-2, [[Exponential distribution|exponential]], and [[Normal distribution|normal]] distributions.
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| Included is a figure showing the relationship between the EGB and its special and limiting cases (see McDonald and Xu (1995) ).
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| [[File:EGBtree.jpg|thumb|EGB distribution tree]]
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| === Moment generating function ===
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| Using similar notation as above, the [[moment-generating function]] of the EGB can be expressed as follows:
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| :<math> M_{EGB}(Z)=\frac{e^{\delta t}B(p+t\sigma,q)}{B(p,q)}{}_{2}F_{1} \begin{bmatrix}
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| p + t\sigma,t\sigma;c \\
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| p + q +t\sigma;
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| \end{bmatrix}.
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| </math>
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| == Uses ==
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| The flexibility provided by the GB family is used in modeling the distribution of:{{cn|date=April 2013}}
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| * family income
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| * stock returns
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| *insurance losses
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| Applications involving members of the EGB family include:{{cn|date=April 2013}}
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| * partially adaptive estimation of regression
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| * time series models
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| ==References==
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| <references /> | |
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| ==Bibliography==
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| * C. Kleiber and S. Kotz (2003) ''Statistical Size Distributions in Economics and Actuarial Sciences''. New York: Wiley
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| * Johnson, N. L., S. Kotz, and N. Balakrishnan (1994) ''Continuous Univariate Distributions''. Vol. 2, Hoboken, NJ: Wiley-Interscience.
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| {{ProbDistributions|continuous-bounded}}
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| [[Category:Continuous distributions]]
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| [[Category:Probability distributions]]
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