Skew-Student’s t Distribution#

Univariate, Continuous, Asymmetric, Unbounded, Heavy-tailed

The skew-Student’s t distribution is a continuous probability distribution that generalizes the Student’s t distribution by allowing for non-zero skewness. It is used in various fields when data is skewed and has heavy tails.

There are several definitions of the skew-Student’s t distribution. Here, we use the definition proposed by Jones and Faddy (2003) that is defined in terms of four parameters: the location parameter (\(\mu\)), the scale parameter (\(\sigma\)), and two shape parameters (\(a\) and \(b\)).

Key properties and parameters#

Support	\(x \in \mathbb{R}\)
Mean	\(\mu + \sigma\frac{(a-b) \sqrt{(a+b)}}{2}\frac{\Gamma\left (a-\frac{1}{2}\right)\Gamma\left(b-\frac{1}{2}\right)}{\Gamma(a) \Gamma(b)}\)
Variance	\(\sigma^2\left(1 + \frac{\Gamma\left(a-\frac{1}{2}\right)\Gamma\left(b-\frac{1}{2}\right)}{\Gamma(a)\Gamma(b)} - \left(\frac{\Gamma\left(a-\frac{1}{2}\right)\Gamma\left(b-\frac{1}{2}\right)}{\Gamma(a)\Gamma(b)}\right)^2\right)\)

Parameters:

\(\mu\) : (float) Location parameter.
\(\sigma\) : (float) Scale parameter.
\(a\) : (float) Shape parameter.
\(b\) : (float) Shape parameter.

Alternative parametrization

The Skew-Student’s t distribution has two alternative parameterizations. In terms of \(\mu\), \(\sigma\), \(a\) and \(b\), or in terms of \(\mu\), \(\lambda\), \(a\) and \(b\).

The link between the two parameterizations is given by:

\[ \lambda = \frac{1}{\sigma^2} \]

If \(a > b\), the skew-Student’s t distribution is positively skewed (skewed to the right). If \(a < b\), the skew-Student’s t distribution is negatively skewed (skewed to the left). If \(a = b\), the skew-Student’s t distribution reduces to the Student’s t distribution with \(\nu = 2a\).

The parameter \(\sigma\) convergences to the standard deviation and \(\lambda\) converges to the precision as \(a\) and \(b\) approach close, and the value of \(a\) gets larger.

Probability Density Function (PDF)#

\[ f(t)=f(t \mid a, b)=C_{a, b}^{-1}\left\{1+\frac{t}{\left(a+b+t^2\right)^{1 / 2}}\right\}^{a+1 / 2}\left\{1-\frac{t}{\left(a+b+t^2\right)^{1 / 2}}\right\}^{b+1 / 2} \]

Where \(C_{a, b}\) is the normalizing constant given by:

\[ C_{a, b}=2^{a+b-1} B(a, b)(a+b)^{1 / 2} \]

and \(B(a, b)\) is the Beta function.

Parameters \(\mu\), \(\sigma\), \(a\), and \(b\)

Parameters \(\mu\), \(\lambda\), \(a\), and \(b\)

Cumulative Distribution Function (CDF)#

\[ F(t \mid a, b) = I_{\frac{1+t/\sqrt{a + b + t^2}}{2}}(a, b) \]

where \(I_x(a, b)\) is the regularized incomplete beta function.