  Envelope method was recently proposed as a method to reduce the dimension of
responses in multivariate regressions. However, when there exists missing data,
the envelope method using the complete case observations may lead to biased and
inefficient results. In this paper, we generalize the envelope estimation when
the predictors and/or the responses are missing at random. Specifically, we
incorporate the envelope structure in the expectation-maximization (EM)
algorithm. As the parameters under the envelope method are not pointwise
identifiable, the EM algorithm for the envelope method was not straightforward
and requires a special decomposition. Our method is guaranteed to be more
efficient, or at least as efficient as, the standard EM algorithm. Moreover,
our method has the potential to outperform the full data MLE. We give
asymptotic properties of our method under both normal and non-normal cases. The
efficiency gain over the standard EM is confirmed in simulation studies and in
an application to the Chronic Renal Insufficiency Cohort (CRIC) study.
