Large sample inference in random coefficient regression models

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Abstract
Random coefficient regression models have been used t odescribe repeated measures on members of a sample of n in dividuals . Previous researchers have proposed methods of estimating the mean parameters of such models. Their methods require that eachindividual be observed under the same settings of independent variablesor , lesss 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 consist ent 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 a11 individuals are observed an equal number of times provided limit 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 i n many bio-medical applications . The primary con tribution 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.

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