  This paper gives a general method for deriving limiting distributions of
complete case statistics for missing data models from corresponding results for
the model where all data are observed. This provides a convenient tool for
obtaining the asymptotic behavior of complete case versions of established full
data methods without lengthy proofs. The methodology is illustrated by
analyzing three inference procedures for partially linear regression models
with responses missing at random. We first show that complete case versions of
asymptotically efficient estimators of the slope parameter for the full model
are efficient, thereby solving the problem of constructing efficient estimators
of the slope parameter for this model. Second, we derive an asymptotically
distribution free test for fitting a normal distribution to the errors.
Finally, we obtain an asymptotically distribution free test for linearity, that
is, for testing that the nonparametric component of these models is a constant.
This test is new both when data are fully observed and when data are missing at
random.
