Measuring Precision of Statistical Inference on Partially Identified Parameters

It has become widely recognized that many types of statistical data only partially identify the parameters of interest as simple as population means, meaning that the parameters cannot be estimated with arbitrary precision simply by increasing the sample size. Statisticians designing surveys and experiments which generate such data could use limited resources either to reduce the extent of partial identi…cation or to reduce sampling error. The former can be accomplished, for example, by putting more e¤ort into pursuing sampled population members who did not respond to a survey. The latter by increasing sample size or improving measurement precision. To inform these choices, it is useful to analytically derive the relative e¤ects of both margins of planning on the precision of inference, which the planner could then compare to their relative costs. The problem was …rst considered in the Cochran-Mosteller-Tukey report on the Kinsey study published in 1954. Concerned with nonrandom nonresponse to the study’s questions, CMT advocated a conservative approach to inference that sets limits on population parameters by allowing for any values of the variable in the part of the population that was not sampled or refused to respond. A variety of applications of the same approach, now known as partial identi…cation, has been developed by Manski (1995, 2007a) and other researchers. CMT calculated for di¤erent sample sizes and refusal rates the relative e¤ects of reducing nonresponse I am grateful to Chuck Manski and Elie Tamer for their helpful comments.