Asymptotic theory for maximum likelihood estimates in reduced-rank multivariate generalised linear models

TL;DR

This paper develops asymptotic theory for maximum likelihood estimators in reduced-rank multivariate generalized linear models, filling a gap in the theoretical understanding of these models.

Contribution

It introduces M-estimation theory for non-convex, non-closed parameter spaces and derives consistency and asymptotic distribution results for reduced-rank estimators.

Findings

Established asymptotic properties of MLE in reduced-rank GLMs

Demonstrated the theory with a real binary classification example

Extended reduced-rank regression theory to more general models

Abstract

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.