  This article considers nonparametric regression models with multivariate
covariates and with responses missing at random. We estimate the regression
function with a local polynomial smoother. The residual-based empirical
distribution function that only uses complete cases, i.e. residuals that can
actually be constructed from the data, is shown to be efficient in the sense of
H\'ajek and Le Cam. In the proofs we derive, more generally, the efficient
influence function for estimating an arbitrary linear functional of the error
distribution; this covers the distribution function as a special case. We also
show that the complete case residual-based empirical distribution function
admits a functional central limit theorem. The article concludes with a small
simulation study investigating the performance of the complete case
residual-based empirical distribution function.
