  Reduced-rank regression is a dimensionality reduction method with many
applications. The asymptotic theory for reduced rank estimators of parameter
matrices in multivariate linear models has been studied extensively. In
contrast, few theoretical results are available for reduced-rank multivariate
generalised linear models. We develop M-estimation theory for concave criterion
functions that are maximised over parameters spaces that are neither convex nor
closed. These results are used to derive the consistency and asymptotic
distribution of maximum likelihood estimators in reduced-rank multivariate
generalised linear models, when the response and predictor vectors have a joint
distribution. We illustrate our results in a real data classification problem
with binary covariates.
