Cheng (1990, 1994) considered a missing data problem in which data comes in two forms, one in which a covariate X is observed, and the other in which both X and a response Y are observed. If the missing data probabilities are independent of X then the distribution of X is the same in the two populations. The goal is to estimate the marginal distribution of Y, and more specifically its mean. Cheng based his estimates on the regression of Y on X in the first population, using parametric and nonparametric regression, and showed that the two methods were roughly comparable in asymptotic efficiency. Motivated by a currently ongoing study, we consider a different problem, namely one in which the two populations are physically distinct in such a way that the distribution of X differs between the populations. We show that the nonparametric modification of Cheng's method appropriate to this situation has zero asymptotic efficiency relative to the parametric approach in a wide class of problems.
