Burr12 Distribution

There are two shape parameters \(c,d > 0\) and the support is \(x \in [0,\infty)\). The Burr12 distribution is also known as the Singh-Maddala distribution.

\begin{eqnarray*} f\left(x;c,d\right) & = & {cd} \frac{x^{c-1}} {\left(1+x^{c}\right)^{d+1}} \\ F\left(x;c,d\right) & = & 1 - \left(1+x^{c}\right)^{-d}\\ G\left(q;c,d\right) & = & \left((1-q)^{-1/d}-1\right)^{-1/c}\\ S\left(x;c,d\right) & = & \left(1+x^{c}\right)^{-d}\\ \mu & = & d \, B\left(d-\frac{1}{c}, 1+\frac{1}{c}\right)\\ \mu_{n} & = & d \, B\left(d-\frac{n}{c}, 1+\frac{n}{c}\right)\\ m_{d} & = & \left(\frac{c-1}{c d + 1}\right)^{1/c} \,\text{if }\quad c>1 \text{, otherwise }\quad 0\\ m_{n} & = & \left(2^{1/d}-1\right)^{-1/c} \end{eqnarray*}

where \(B(a, b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}\) is the Beta function.

Implementation: scipy.stats.burr12