This research includes statistical inference in models in which the multivariate observation is treated as the sum of a 'systematic part' and a random 'error'. The systematic part, which is stochastic in the multivariate components of variance model and may be stochastic, nonstochastic, or mixed in the factor analysis models, is in many cases restricted to a linear space of lower dimension than the observation and error. The covariance matrix of the error may have many special properties; for example, in factor analysis it is diagonal. This research is in the general area of statistics. The models studied in the research are used extensively in psychometrics and econometrics. In the research large-sample theory will be developed for tests of hypotheses concerning the number of linear restrictions and procedures to determine the number of linear restrictions, and for the estimators of the restrictions and the covariance matrices.
