doc/source/tutorial/stats/continuous_jf_skew_t.rst
.. _continuous-jf_skew_t:
A skew extension of the t distribution, defined for :math:a>0 and
:math:b>0.
.. math:: :nowrap:
\begin{eqnarray*}
f(x;a,b) & = & C_{a,b}^{-1} \left(1+\frac{x}{\left(a+b+x^2\right)^{1/2}}\right)^{a+1/2} \left(1-\frac{x}{\left(a+b+x^2\right)^{1/2}}\right)^{b+1/2} \\
F(x;a,b) & = & I\left(\frac{1+x(a+b+x^2)^{-1/2}}{2};a,b\right) \\
\mu_{n}^{\prime} & = & \frac{(a+b)^{n/2}}{2^nB(a,b)}\sum_{i=0}^{n}{n \choose i}(-1)^iB\left(a+\frac{n}{2}-i, b-\frac{n}{2}+i\right)
\end{eqnarray*}
where :math:C_{a,b}=2^{a+b-1}B(a,b)(a+b)^{1/2}, :math:B is the beta
function scipy.special.beta and the formula for the moments
:math:\mu_{n}^{\prime} holds provided that :math:a>n/2 and :math:b>n/2.
When :math:a<b, the distribution is negatively skewed, and when :math:a>b,
the distribution is positively skewed. If :math:a=b, then we recover the t
distribution with :math:2a degrees of freedom.
10.1111/1467-9868.00378Implementation: scipy.stats.jf_skew_t