Consider an inference problem on a parameter p=k+l, where k is a point estimate of normally distributed data and l is an partially identified interval (due to survey nonresponse, for instance). Should a researcher spend more resources reducing the standard error of k (by increasing sample size), or reducing the interval of partial identification (by tracking down nonrespondents)? This turns out to hinge on the measure desired. Tetenov reports optimal strategies if the researcher is reporting a 95% confidence interval for p, or if the researcher is trying to minimize the maximum mean square error, maximum mean absolute deviation or maximum regret. The point results all suggest placing more emphasis on minimizing the size of the partially identified interval compared to the optimal strategy when the confidence interval is reported.
“Measuring Precision of Statistical Inference on Partially Identified Parameters,” A. Tetenov (2009)