Acceptance-Rejection

If we denote the indicator function by ${\bf I}(e)$ (with ${\bf I}(e)=1$ if $e$ is true; otherwise ${\bf I}(e)=0$), then a simple acceptance-rejection (AR) algorithm for the MVT problem, using equation (7), uses

$${\bf T}({\bf a}, {\bf b}, \mbox{\boldmath$\Sigma$}, \nu) \ap... ...ts_{k=1}^N{\bf I}({\bf a}\leq C{\bf y}({\bf u}_k)\leq{\bf b}),$$ (8)

where $\{{\bf u}_k\}$ is random with components $u_{i,k} \sim t_{\nu+i-1}$, and we define the univariate t-distribution function by $t_\nu(u) = K_\nu^{(1)} \displaystyle\int \limits_{-\infty}^{u}(1+\frac{s^2}{\nu})^{-\frac{1+\nu}{2}}ds$.

A simple AR algorithm for the NCMVT problem, based on equation (6), uses

$${\bf T}({\bf a}, {\bf b}, \mbox{\boldmath $\Sigma$}, \nu, \m... ...f y}_k \leq \frac{s_k{\bf b}}{\nu}-\mbox{\boldmath $\delta$}),$$

where $\{{\bf y}_k\}$ is random with components $y_{i,k}\sim N(0,1)$, and where $\{s_k\}$ is random with $s_k\sim \chi_\nu$, and we define $\chi_\nu(u) =\frac{2^{1-\frac{\nu}{2}}}{\Gamma(\frac{\nu}{2})} \int\limits_0^us^{\nu-1}e^{-\frac{s^2}{2}}ds$.