owens_t() [4/8]

double stan::math::owens_t ( double  h, double  a  )

inline

Return the result of applying Owen's T function to the specified arguments.

Used to compute the cumulative density function for the skew normal distribution.

\[ \mbox{owens\_t}(h, a) = \begin{cases} \mbox{owens\_t}(h, a) & \mbox{if } -\infty\leq h, a \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } h = \textrm{NaN or } a = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{owens\_t}(h, a)}{\partial h} = \begin{cases} \frac{\partial\, \mbox{owens\_t}(h, a)}{\partial h} & \mbox{if } -\infty\leq h, a\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } h = \textrm{NaN or } a = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{owens\_t}(h, a)}{\partial a} = \begin{cases} \frac{\partial\, \mbox{owens\_t}(h, a)}{\partial a} & \mbox{if } -\infty\leq h, a\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } h = \textrm{NaN or } a = \textrm{NaN} \end{cases} \]

\[ \mbox{owens\_t}(h, a) = \frac{1}{2\pi} \int_0^a \frac{\exp(-\frac{1}{2}h^2(1+x^2))}{1+x^2}dx \]

\[ \frac{\partial \, \mbox{owens\_t}(h, a)}{\partial h} = -\frac{1}{2\sqrt{2\pi}} \operatorname{erf}\left(\frac{ha}{\sqrt{2}}\right) \exp\left(-\frac{h^2}{2}\right) \]

\[ \frac{\partial \, \mbox{owens\_t}(h, a)}{\partial a} = \frac{\exp\left(-\frac{1}{2}h^2(1+a^2)\right)}{2\pi (1+a^2)} \]

Parameters

h	First argument
a	Second argument

Returns

Owen's T function applied to the arguments.

Definition at line 58 of file owens_t.hpp.