Large sample inference in random coefficient regression models

Random coefficient regression models have been used to describe repeated measures on members of a sample of n individuals, Previous researchers have proposed methods of estimating the mean parameters of such models. Their methods require that each individual be observed under the same settings of independent variables or, less stringently, that the number of observations, r, on each individual be the same. Under the latter restriction, estimators of mean regression parameters exist which are consistent as both r^➛∞and n^➛∞, and efficient as r^➛∞and large sample (r large) tests of mean parameters are available. These results are easily extended to the case where not all individuals are observed an equal number of times provided limits are taken as min(r) ➛∞, Existing methods of inference, however, are not justified by the current literature when n is large and r is small, as is the case in many bio-medical applications. The primary contribution of the current paper is a derivation of the asymptotic properties of modifications of existing estimators as n alone tends to infinity, r fixed. From these properties it is shown that existing methods of inference, which are currently justified only when min(r) is large, are also justifiable when n is large and min(r) is small. A secondary contribution is the definition of a positive definite estimator of the covariance matrix for the random coefficients in these models. Use of this estimator avoids computational problems that can otherwise arise.