We propose inference procedures for partially identified population features for which thepopulation identification region can be written as a transformation of the Aumann expectationof a properly defined set valued random variable (SVRV). An SVRV is a mapping that associatesa set (rather than a real number) with each element of the sample space. Examples ofpopulation features in this class include sample means and best linear predictors with intervaloutcome data, and parameters of semiparametric binary models with interval regressor data.We extend the analogy principle to SVRVs, and show that the sample analog estimator of thepopulation identification region is given by a transformation of a Minkowski average of SVRVs.Using the results of the mathematics literature on SVRVs, we show that this estimator convergesin probability to the identification region of the model with respect to the Hausdorffdistance. We then show that the Hausdorff distance between the estimator and the populationidentification region, when properly normalized by √n, converges in distribution to the supremumof a Gaussian process whose covariance kernel depends on parameters of the populationidentification region. We provide consistent bootstrap procedures to approximate this limitingdistribution. Using similar arguments as those applied for vector valued random variables, wedevelop a methodology to test assumptions about the true identification region and to calculatethe power of the test. We show that these results can be used to construct a confidence collection,that is a collection of sets that, when specified as null hypothesis for the true value of thepopulation identification region, cannot be rejected by our test.
Centre for microdata methods and practice