MANOVA with singular variance matrix

Abstract

Classical multivariate analysis of variance for p response variables is extended to cover high-dimensional data. For example, data often comprise many response variables that may be related. Therefore, inference based on all the response variables may be inefficient. However, the relationship between the response variables is usually not known. This leads to the assumption that the p response variables span a linear space of some fixed dimension, say r<p; equivalently the p×p variance matrix is singular of rank r. We will assume that the rank is given. Following the classical approach of doing inference in linear models, parameters are first estimated and thereafter tests are constructed. Estimators and tests are based on the likelihood method. The present model differs from the classical multivariate analysis of variance model and consists of a deterministic and a random part. It is noticed that the classical approach is a special case of the one which will be considered in this article.