If you have two Bernoulli random variables, $X$ and $Y$ with success probabilities $a$ and $b$, both independent of each other, and we define a third random variable $Z = X+Y$, is it possible to recover $min\{a,b\}$ and $max\{a,b\}$ from samples from Z (i.e $Z_{1}, Z_{2}, ..., Z_{n}$)?
I know that it isn't possible to recover either a or b, since Z is defined by the sum of a and b, so I'm lead to believe that you couldn't recover the min or max as doing so would effectively identify/recover a and b.
$Z$ takes values $0,1,2$ with respective probabilities $(1-a)(1-b), a(1-b)+(1-a)b, ab$.
Given i.i.d. samples $Z_1, \ldots, Z_n$, the log likelihood is $$N_0 \log((1-a)(1-b)) + N_1 \log(a(1-b) + (1-a)b) + N_2 \log(ab)$$ where $N_i = \# \{k : Z_k = i\}$.
In principle you can try to choose $a$ and $b$ to maximize this likelihood, but I am not sure if this is a concave function, and taking derivatives is a little messy. Maybe you or someone else can try digging further.