# Rejection-Samplig starting from Cumulative Distributions

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: $$\begin{equation*} f_X(x) \le M f_Y(x) \end{equation*}$$