  The assumption that response and predictor belong to the same statistical
unit may be violated in practice. Unbiased estimation and recovery of true
label ordering based on unlabeled data are challenging tasks and have attracted
increasing attentions in the recent literature. In this paper, we present a
relatively complete analysis of label permutation problem for the generalized
linear model with multivariate responses. The theory is established under
different scenarios, with knowledge of true parameters, with partial knowledge
of underlying label permutation matrix and without any knowledge. Our results
remove the stringent conditions required by the current literature and are
further extended to the missing observation setting which has never been
considered in the field of label permutation problem. On computational side, we
propose two methods, "maximum likelihood estimation" algorithm and "two-step
estimation" algorithm, to accommodate for different settings. When the
proportion of permuted labels is moderate, both methods work effectively.
Multiple numerical experiments are provided and corroborate our theoretical
findings.
