User:Fiarisk/sandbox

In probability and statistics, the Jones and Faddy Skew t (JFST ***) 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

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 $$ab$$, 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.

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


 * $$E[T^r]=\frac{(a+b)^{r/2}}{2^rB(a,b)}\sum_{i=0}^{r}{r \choose i}(-1)^iB\left(a+\frac{r}{2}-i, b-\frac{r}{2}+i\right),$$

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


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