Extreme value estimation for a function of a random sample using binomial moments scheme and Boolean functions of events (Q505450)

The authors consider the following problem. Given function \(f\) of \(n\) variables and a random sample \(X = (X_1, \dots, X_n)\) find lower an upper bounds, \(l\) and \(u\), such that \(P(l \leq f(X) \leq u) \simeq 1\). They note that these numbers might be obtained by solving two optimization problems: \(\min f(x)\) subject to \(x \in \Omega\), \(\max f(x)\) subject to \(x \in \Omega\), where \(\Omega\) is the set of all possible realizations of \(X\). Them they show a method to deal with them based on the binomial moments to construct a systematic mathematical form, and Boolean logic.