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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!

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