Consider two independent random Variables $X, Y$, and $X>=0, Y>=0$. $f(X, Y)$ is monotonic to $X, Y$ respectively.
Suppose that we know the 0%, 25%, 50%, 75%, 100% quantiles for both $X, Y$ and we have access to $f(\cdot, \cdot)$.
Is it possible to find some quantiles of the new R.V. $Z = f(X, Y)$?
Quantile is invariant to monotonic transformation but is there any nice property when we consider the scenario above?
I appreciate a lot for any hint. :)
Well, if you know $100\%$ quantiles, i.e. you know $c$ and $d$ such that $X \le c$ and $Y \le d$ with probability $1$, then $Z = f(X,Y) \le f(c,d)$ with probability $1$. This says $f(c,d)$ is a $100\%$ quantile for $Z$. (I say a rather than the, because in general quantiles are not unique).
Similarly for $0\%$.
Not for any of the others, though: they would require knowing the actual distributions, at least in some interval.