Inverse set estimation and inversion of simultaneous confidence intervals
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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 in the estimation of such a set, we propose data-dependent inner and outer confidence sets that are sub- and super-sets of the true inverse set with a given confidence. Existing methods require strict assumptions, and the predefined subset of the range is usually an excursion set for only one single level. We show that by inverting pre-constructed simultaneous confidence intervals, commonly available for different kinds of data, multiple confidence sets of multiple levels can be simultaneously constructed with the desired confidence non-asymptotically. The method is illustrated on dense functional data to determine regions with rising temperatures in North America and on logistic regression data to assess the effect of statin and COVID-19 on clinical outcomes of in-patients.
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