< Probability distributions < Generalized beta distribution

The **generalized Beta distribution (GB)** is an extension of the beta distribution, a continuous probability distribution characterized by two shape parameters. The GB, with up to five parameters, can model a broader range of data in comparison to the beta distribution. It is frequently used in Bayesian statistics and regression analysis as well as practical applications such as modeling income distribution and stock returns.

The GB’s flexibility makes it suitable for modeling many different types of data, including unimodal, increasing, decreasing, and bath-tub shaped distributions depending on the parameter values [1].

## Generalized beta distribution properties

Many forms of the probability density function (pdf) exist. The following form is “dimensionless” [2], which means that the distribution doesn’t depend on any specific units of measurement. The result is that this form of the pdf can be used to model a variety of data, without having to convert the data to a common unit of measurement:

where

- β
_{1}and β_{2}are scale parameters, - α, p and q are shape parameters, all positive,
- B(p, q) is the beta function and x ≤ β
_{1}.

The corresponding cumulative distribution function (CDF) is [3]:

where

- I(y; p, q) = B(y; p, q)/B(p, q) is the regularized beta function and
- B(y; p, q) is the incomplete beta function.

## The GB Family

The GB encompasses over thirty named distributions as limiting or special cases. The exponential generalized beta (EGB) distribution directly follows from the GB and extends other common distributions.

The three four parameter distributions depicted in the tree are the generalized beta of the first kind (GB1), the beta distribution, and the generalized beta of the second kind (GB2, also called the *generalized beta prime*) [5]. The generalized beta distribution of the second kind was used by Thurow in 1970 [6] for income distribution; over a decade later McDonald [7] used the GB of the first and second kind for the same purpose, finding that the performance of the GB1 is comparable to the generalized gamma distribution and beta of the second kind distributions.

The GB1 includes the following distributions:

- B1 = beta of the first kind (B1),
- GG = generalized gamma distribution
- P = Pareto type 1 distribution.

It also includes the following distributions

- Generalized Power distribution
- Kumaraswamy distribution
- Stoppa distribution ((generalized Pareto type 1, not depicted)

The GB2 is useful for unifying a substantial number of distributions including special cases:

- GG =generalized gamma distribution
- B2 = beta distribution of second kind
- BR3 = Burr III distribution
- BR12 =Burr XII distribution
- Fisk = log-logistic distribution
- L = Lomax distribution
- IL = inverse Lomax distribution
- GA = gamma distribution
- W = Weibull distribution
- EXP = exponential distribution.

It is not usually recommended to use the GB2 family with small sample sizes (fewer than several hundred) as numerical problems easily occur with small samples [8].

## References

[1] Ng, D. et al. The study of properties on generalized Beta distribution. 3rd International Conference on Mathematical Sciences and Statistics. IOP Conf. Series: Journal of Physics: Conf. Series 1132 (2018) 012080IOP Conf. Series: Journal of Physics: Conf. Series 1132 (2018) 012080

[2] J. B. McDonald, Some generalized functions for the size distribution of income, Econometrica 52 (3) (1984) 647–665.

[3] J. B. McDonald, Y. J. Xu, A generlazition of the beta distributionwith applications, Journal of Econometrics 66 (1996)

133–152. As cited in Liu & Serota. Rethinking Generalized Beta Family of Distributions. Retrieved June 8, 2024 from https://arxiv.org/pdf/2209.05225

[4] Dmauler, CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons

[5] 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.

[6] Thurow, L. C. (1970). Analyzing the American Income Distribution. Papers and

proceedings, American Economics Association, 60, 261-269

[7] McDonald, J.B., (1984), Some generalized functions for the size distribution of

income. Econometrica, 52, 647-664

[8] Yees, T. & Miranda, V. genbetaII: Generalized Beta Distribution of the Second Kind. Retrieved August 14, 2021 from: https://rdrr.io/cran/VGAM/man/genbetaII.html