  Many partial identification problems can be characterized by the optimal
value of a function over a set where both the function and set need to be
estimated by empirical data. Despite some progress for convex problems,
statistical inference in this general setting remains to be developed. To
address this, we derive an asymptotically valid confidence interval for the
optimal value through an appropriate relaxation of the estimated set. We then
apply this general result to the problem of selection bias in population-based
cohort studies. We show that existing sensitivity analyses, which are often
conservative and difficult to implement, can be formulated in our framework and
made significantly more informative via auxiliary information on the
population. We conduct a simulation study to evaluate the finite sample
performance of our inference procedure and conclude with a substantive
motivating example on the causal effect of education on income in the
highly-selected UK Biobank cohort. We demonstrate that our method can produce
informative bounds using plausible population-level auxiliary constraints. We
implement this method in the R package selectioninterval.
