User:Fiarisk/sandbox

Jones and Faddy Skew t

Parameters	$a\in \mathbb {R} _{>0}$ $b\in \mathbb {R} _{>0}$
Support	$t\in (-\infty ,\infty )$
PDF	$C_{a,b}^{-1}\left(1+{\frac {t}{\left(a+b+t^{2}\right)^{1/2}}}\right)^{a+1/2}\left(1-{\frac {t}{\left(a+b+t^{2}\right)^{1/2}}}\right)^{b+1/2},$ where $C_{a,b}=2^{a+b-1}B(a,b)(a+b)^{1/2}$ and $B$ denotes the beta function
CDF	$I_{g(t;a,b)}(a,b),$ where $g(t;a,b)={\frac {1+t(a+b+t^{2})^{-1/2}}{2}}$ and $I_{x}$ denotes the regularized incomplete beta function
Mean	${\frac {(a-b)(a+b)^{1/2}}{2}}{\frac {\Gamma (a-{\frac {1}{2}})\Gamma (b-{\frac {1}{2}})}{\Gamma (a)\Gamma (b)}}$ for $a>{\frac {1}{2}}$ and $b>{\frac {1}{2}}$
Mode	${\frac {(a-b)(a+b)^{1/2}}{{\sqrt {2a+1}}{\sqrt {2b+1}}}}$

In probability and statistics, the Jones and Faddy Skew t (JFST ***) [1] distribution is an extension of Student's t-distribution (t-distribution) which allows for skewness in addition to the heavy-tails allowed for by the t-distribution. It includes the t-distribution as a special case.

https://www.pp.rhul.ac.uk/~cowan/stat/skew_t_jones_and_faddy.pdf

Definition

Probability density function

For parameters $a>0$ and $b>0$, the JFST distribution has the probability density function (PDF)

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

where

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

and $B$ denotes the beta function.

When $a<b$, the distribution is negatively skewed, and when $a>b$, the distribution is positively skewed. When $a$ is equal to $b$, $f$ reduces to the PDF of the t-distribution on $2a$ degrees of freedom.

Cumulative distribution function

For parameters $a>0$ and $b>0$, the JFST distribution has the cumulative density function (CDF)

$F(t)=F(t;a,b)=I_{g(t;a,b)}(a,b),$

where

$g(t;a,b)={\frac {1+t(a+b+t^{2})^{-1/2}}{2}}$

and $I_{x}$ denotes the regularized incomplete beta function.

Properties

Moments

For $a>r/2$ and $b>r/2$, the $r$^th raw moment of the JFST distribution is

$E[T^{r}]={\frac {(a+b)^{r/2}}{2^{r}B(a,b)}}\sum _{i=0}^{r}{r \choose i}(-1)^{i}B\left(a+{\frac {r}{2}}-i,b-{\frac {r}{2}}+i\right),$

where $B$ denotes the beta function.

The expected value of a JFST distribution with parameters $a>1/2$ and $b>1/2$ is

$E[T]={\frac {(a-b)(a+b)^{1/2}}{2}}{\frac {\Gamma (a-{\frac {1}{2}})\Gamma (b-{\frac {1}{2}})}{\Gamma (a)\Gamma (b)}},$

where $\Gamma$ is the gamma function, and the second moment, for $a>1$ and $b>1$, is

$E[T^{2}]={\frac {(a+b)}{4}}{\frac {(a-b)^{2}+a-1+b-1}{(a-1)(b-1)}}$.

Mode

The PDF $f$ is unimodal, with the mode equal to

${\frac {(a-b)(a+b)^{1/2}}{{\sqrt {2a+1}}{\sqrt {2b+1}}}}$.