I am interested to determine if there are explicit formulas for the parameters of the Generalized Beta (GB) distribution. McDonald [1] eludes to this but does not provide solution. I was wondering if someone has run into this before or has an idea of how to proceed. My work is below the description and set up.
The General Beta distribution is defined as the following:
\begin{equation} GB(y;a,b,c,p,q) = \frac{|a| y^{ap-1}(1-(1-c)\left(\frac{y}{b})\right)^{q-1}}{b^{ap}B(p,q)(1+c\left(\frac{y}{b}\right)^a)^{p+q}} \end{equation}
where $b,p, q > 0$, $0\le c \le 1$, and $0 < y^a < \frac{b^a}{1-c}$, and $B(p,q)$ is the Beta function of $p$ and $q$.
McDonald [1] show that the moments of the GB distribution are:
\begin{equation} \mathbb{E}_{GB}(Y^h)=\frac{b^hB\left(p+\frac{h}{a},q\right)}{B(p,q)} {_{2}F_1}\left(p+\frac{h}{a}, \frac{h}{a}; c; p+q+\frac{h}{a}\right) \end{equation}
where ${_{2}F_1}$ is the hypergeometric series.
Current Efforts:
The corresponding probability density function for a sample of $n$ independent identically distributed (iid) GB random variables is:
\begin{equation} f(y_1,y_2,...,y_n|a,b,c,p,q) = \prod_{j=1}^n \frac{|a| y_j^{ap-1}\left(1-(1-c)\left(\frac{y_j}{b}\right)\right)^{q-1}}{b^{ap}B(p,q)\left(1+c\left(\frac{y_j}{b}\right)^a\right)^{p+q}} \end{equation}
This family of distributions has 5 parameters: $\theta = (a,b,c,p,q)$; so we maximize the likelihood, ${\displaystyle {\mathcal {L}}(a,b,c,p,q )=f(y_{1},\ldots ,y_{n}\mid a,b,c,p,q)}$ over all five parameters simultaneously, or if possible, individually.
Since the logarithm function itself is a continuous strictly increasing function over the range of the likelihood, the values which maximize the likelihood will also maximize its logarithm (the log-likelihood itself is not necessarily strictly increasing). The log-likelihood can be written as follows:
\begin{equation} \ln\left({\mathcal {L}}(a,b,c,p,q )\right) = \ln\left(\prod_{j=1}^n \frac{|a| y_j^{ap-1}\left(1-(1-c)\left(\frac{y_j}{b}\right)\right)^{q-1}}{b^{ap}B(p,q)\left(1+c\left(\frac{y_j}{b}\right)^a\right)^{p+q}}\right) \end{equation}
and becomes
\begin{equation} \ln\left({\mathcal {L}}\right) = n\ln a +(ap-1)\sum_{j=1}^n \ln(y_j) + (q-1)\sum_{j=1}^n \ln\left(1-(1-c)\left(\frac{y_j}{b}\right)^a\right) - apn \ln b - n \ln B(p,q) (p+q) \sum_{j=1}^n \ln\left(\left(1+c\frac{y_j}{b}\right)^q\right) \end{equation}
Computing the derivatives of the log-likelihood, we obtain
\begin{equation} \frac{\partial {\mathcal {L}}}{\partial a} = \frac{n}{a} - p n \ln b + p \sum_{j=1}^n \ln y_j - \sum_{j=1}^n \frac{(1-c)(q-1)\left(\frac{y}{b}\right)^a \ln\left(\frac{y_j}{b}\right)}{1-(1-c)\left(\frac{y_j}{b}\right)^a} \end{equation}
\begin{equation} \frac{\partial {\mathcal {L}}}{\partial b} = \frac{a n p}{b} + \sum_{j=1}^n \frac{a(1-c)(q-1)y_j \left(\frac{y_j}{b}\right)^{a-1}}{b^2 \left((1-(1-c)\left(\frac{y_j}{b}\right)^a\right)} + \sum_{j=1}^n \frac{c q (p+q) y_j \left(\frac{y_j}{b}\right)^{q-1}}{b^2\left(1+c\left(\frac{y_j}{b}\right)\right)} \end{equation}
\begin{equation} \frac{\partial {\mathcal {L}}}{\partial c} = \sum_{j=1}^n \left[ \frac{(q-1)\left(\frac{y_j}{b}\right)^a}{1-(1-c)\left(\frac{y_j}{b}\right)^a} -\frac{(p+q)\left(\frac{y_j}{b}\right)^q}{1+c\left(\frac{y_j}{b}\right)^q}\right] \end{equation}
\begin{equation} \frac{\partial {\mathcal {L}}}{\partial p} = -a n \ln b +\sum_{j=1}^n \left[ a \ln y_j - \ln \left(1+c\left(\frac{y_j}{b}\right)^q\right)^q - \psi^{(0)}(p) + \psi^{(0)}(p+q) \right] \end{equation}
\begin{equation} \frac{\partial {\mathcal {L}}}{\partial q} = \frac{n}{a} + p \sum_{j=1}^n \left[\ln\frac{y_j}{b} - \frac{(1-c)(q-1)\left(\frac{y_j}{b}\right)^a \ln \frac{y_j}{b}}{1-(1-c)\left(\frac{y_j}{b}\right)^a}\right] \end{equation}
where $\psi$ represents the polygamma function.
I have noticed that for certain cases I can solve for one of the parameters in tersm of the others but I was unable to find an approach to solve for each explicitly in terms of the observations only. For example, if we look at $\frac{\partial {\mathcal {L}}}{\partial a}$ then we notice that we can solve for $p$ in terms of the other parameters.
I am not certain that this approach would let me solve for these parameters explicitly - or if it is even possible to solve for them explicitly. Any assistiance would be most appreciative.
[1] 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
