I recently asked this question (link here) regarding generating samples of a Gaussian vector with independent components above a hyperplane. Before getting an answer, I had already performed a simple simulation in 2-D: Let $Z=(Z_1,Z_2)$ be a Gaussian vector with $N(0,1)$ independent components. I tried generating samples of $Z$ such that $Z_1>4$ (i.e. my hyperplane) using what I thought was an intuitive approach:
1) Generate a sample of $Z_2 \sim N(0,1)$.
2) Since $Z_1,Z_2$ are independent, I just had to generate a sample of $Z_1 > 4$.
3) Then this gives me a sample of $Z = (Z_1,Z_2)$.
Now, I am interested in generating samples of a Gaussian vector that lie above a hypersurface. For simplicity, let's consider the 2-D example in which I want to generate $Z$ (same vector above) such that $Z_1 > Z_2^2 + 3$. I adapted the "intuitive" approach I pursued above as follows:
1) Generate a sample of $Z_2 \sim N(0,1)$ which we denote by $z_2$.$
2) Then generate a sample of $Z_1$ that satisfies $Z_1 > z_2^2 + 3$.
I compared the samples I obtained from the above approach with what I would get from standard rejection sampling. However, I realized that the above approach is wrong.
I am trying to understand why the method I described above fails for the hypersurface I tried, but I don't see it intuitively. Since the above approach fails for the hypersurface I tried, is there a general efficient approach to generate samples of a Gaussian vector with independent components above a hypersurface assuming that the functional form of this hypersurface is known? Note that I'm ruling out standard rejection sampling because it is highly inefficient for rare events.
Thoughts on either question (especially why the above approach fails) greatly appreciated.