This dissertation is concerned with sets of random variables connected by relationships which may be interpreted as exact linear relations between unobserved 'true' values which are obscured by random 'errors' or 'departures'. The great majority of published work on this topic relates to two random variables connected by a single relation; examples are given here in which it is necessary to consider more than two variates and more than one relation between them. Methods are proposed for representing such relationships between an arbitrary number of random variables, and connexions are established with other statistical models, in particular factor analysis. After a review of methods of estimation proposed for certain cases of varying generality, a fairly comprehensive treatment is given of the estimation of such relationship by the methods of maximum likelihood and generalized least-squares; the large-sample behaviour of the estimators is considered, and connexions with the technique of canonical analysis are established. In the course of this study, some inequalities for matrix traces are derived which are of wider mathematical interest. One of the procedures developed here is applied to a problem of comparing different instruments designed to measure the same property, examining their relative calibrations and relative precision. The data are also used to illustrate graphical techniques developed for testing the assumptions of the calibration model.
|Date of Award||1975|