Jones and Faddy Skew-T Distribution

A skew extension of the t distribution, defined for \(a>0\) and \(b>0\).

\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 \(C_{a,b}=2^{a+b-1}B(a,b)(a+b)^{1/2}\), \(B\) is the beta function scipy.special.beta and the formula for the moments \(\mu_{n}^{\prime}\) holds provided that \(a>n/2\) and \(b>n/2\).

When \(a<b\), the distribution is negatively skewed, and when \(a>b\), the distribution is positively skewed. If \(a=b\), then we recover the t distribution with \(2a\) degrees of freedom.

References

• M.C. Jones and M.J. Faddy. “A skew extension of the t distribution, with applications” Journal of the Royal Statistical Society, Series B (Statistical Methodology) 65, no. 1 (2003): 159-174. DOI:10.1111/1467-9868.00378

Implementation: scipy.stats.jf_skew_t