Inverse set estimation and inversion of simultaneous confidence intervals

The preimage or inverse image of a predefined subset of the range of a deterministic function, called inverse set for short, is the set in the domain whose image equals that predefined subset. To quantify the uncertainty present in estimating such a set, one can construct data-dependent inner and outer confidence sets that serve as sub- and super-sets respectively of the true inverse set. Existing methods require strict assumptions with emphasis on dense functional data. In this work, we generalize the estimation of inverse sets to wider range data types by rigorously proving that, by inverting pre-constructed simultaneous confidence intervals (SCI), confidence sets of multiple levels can be simultaneously constructed with the desired confidence non-asymptotically. We provide valid non-parametric bootstrap algorithm and open source code for constructing confidence sets on dense functional data and multiple regression data. The method is exemplified in two distinct applications: identifying regions in North America experiencing rising temperatures using dense functional data and evaluating the impact of statin usage and COVID-19 on the clinical outcomes of hospitalized patients using logistic regression data.