  A prevailing challenge in the biomedical and social sciences is to estimate a
population mean from a sample obtained with unknown selection probabilities.
Using a well-known ratio estimator, Aronow and Lee (2013) proposed a method for
partial identification of the mean by allowing the unknown selection
probabilities to vary arbitrarily between two fixed extreme values. In this
paper, we show how to leverage auxiliary shape constraints on the population
outcome distribution, such as symmetry or log-concavity, to obtain tighter
bounds on the population mean. We use this method to estimate the performance
of Aymara students---an ethnic minority in the north of Chile---in a national
educational standardized test. We implement this method in the new statistical
software package scbounds for R.
