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Many partial identification problems can be characterized by the optimalvalue of a function over a set where both the function and set need to beestimated by empirical data. Despite some progress for convex problems,statistical inference in this general setting remains to be developed. Toaddress this, we derive an asymptotically valid confidence interval for theoptimal value through an appropriate relaxation of the estimated set. We thenapply this general result to the problem of selection bias in population-basedcohort studies. We show that existing sensitivity analyses, which are oftenconservative and difficult to implement, can be formulated in our framework andmade significantly more informative via auxiliary information on thepopulation. We conduct a simulation study to evaluate the finite sampleperformance of our inference procedure and conclude with a substantivemotivating example on the causal effect of education on income in thehighly-selected UK Biobank cohort. We demonstrate that our method can produceinformative bounds using plausible population-level auxiliary constraints. Weimplement this method in the R package selectioninterval.

Sample-constrained partial identification with application to selection bias