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.
This article was first posted without an Acknowledgements section. The Acknowledgements section was added on 19 October 2021.
The authors were supported by NSF DMS 1916013 and NIH U24-DK-060990. The authors sincerely thank professor Dennis R. Cook, the editor, an associate editor, and a reviewer for their constructive comments that led to a much improved paper.
"Envelope method with ignorable missing data." Electron. J. Statist. 15 (2) 4420 - 4461, 2021. https://doi.org/10.1214/21-EJS1881