Approximate distribution of a complicated function of a random variable If $X$ is a random variable cdf $F(x)$ such that $F$ is invertible then we have the standard method of finding the pdf of any function of $X$, say,  $\sin(X) $  or $ X^3+1 $.However,in many situations we have very complicated composite  functions of random variables(with known distributions ,however).To take an example ,let $X_1,X_2,\dots,X_n$ be iid random variables following uniform distributions in the interval (0,1).Consider the following random varibales
$$ Y=\max_{1 \leq i
\leq n}X_i$$
$$ Z= \frac{1}{2}\left\{ 1+X_{n-1}+ \sqrt{\left(1-X_{n-1}\right)^2+4Y} \right\}$$
To the best of my knowledge, it is almost impossible to analytically find the distribution of the  $Z$. However, it is straight-forward to numerically simulate $Z$.I want to know if there are some methods of finding the approximate distributions of random variables defined like $Z$. Any references/links/hints/suggestions will be greatly appreciated. I believe the problem is of immense significance in the theory of probability and machine learning and other applied disciplines, but I honestly do not know much about it.

You can approximate the $\max(\cdot)$ with a smooth function. See R. Vershynin's book, the proof of the Gaussian comparison inequalities such as the Slepian's inequality.
Then you can use the multivariate change of variable formula for the multivariate smooth transformation, say, $(X_1,\dots,X_{n}) \mapsto (X_1,\dots,X_{n-1},Z)$. This is well known and involves the determinant of the Jacobian, assuming that you have densities w.r.t. the Lebesgue measure.
You may also be interested in the Area and Coarea formulas (geometric measure theory) which are rigorous ways of expressing the above transformation of the densities.
These are general ideas. For the specific example above, the are other methods (say writing $\max_{1 \le i \le n} X_i = \max\{X_{n-1}, V\}$ where $V = \max\{X_j, j \neq n-1\}$ as pointed out in the other answer.
