Could you please help me on the following problem on a rejection sampling method?

I know the cdf of a random variables $X$ that depends on two positive parameters $a\ge 0$, $b\ge 0$ with $a\le b$, denoted $F_X(x, a, b)$.

I can write $F_X(x, a, b)$ in terms of the cdf of a standard Gaussian $\Phi(x)$ as follows \begin{equation*} F_X(x) = \Phi\left(h_1(x, a, b)\right) + \left(1-\frac{2a}{b}\right)h_2( a, b) \Phi\left(h_3(x, a, b)\right) \end{equation*} where $h_1$, $h_2$, $h_3$ are well defined functions in particular $h_2( a, b)\ge 1$ is independent of $x$.

I would like to find a way to simulate $X$ and I am only interested in the case $a\ge b/2$.

If $a = b/2$ the second terms disappears and I can also invert $\Phi\left(h_1(x, a, b)\right)$ therefore I thought for $a>b/2$ I could set-up a rejection algorithm.

Indeed in this case I can write \begin{equation*} F_X(x) \le \Phi\left(h_1(x, a, b)\right) \end{equation*}

Once more I can invert the right hand side and therefore simulate the random variable $Y$ whose cdf is $F_Y(x) = \Phi\left(h_1(x, a, b)\right)$. Of course I also know the density $f_Y(x)$ of $Y$.

Probably I am too naive or on the wrong direction, do you have any idea on how (and if) I could the set-up a rejection-sampling method such that: \begin{equation*} f_X(x) \le M f_Y(x) \end{equation*}

Many thanks in advance for your help!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.