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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