The Mixed Effects Model and Simultaneous Diagonalization of Symmetric Matrices

Abstract

This paper presents results obtained in efforts to generalize ideas reported during the past 15 years on the subject of variance components. The paper applies the result of Newcomb (1960) to the mixed effects model $Y = \sum^t_{j=1} X_{j\tau j} + \sum^h_{k=1} Z_kb_k$ where the $\tau_j$ are fixed parameters and the $b_k$ are random vectors distributed normally with covariance matrix $\sigma k^2I$. The restrictions made on the model are far less severe than those imposed by other authors. Through a nonsingular transformation of the data vector $Y$, minimal sufficient statistics are obtained. Theorems are presented which give conditions under which minimum variance unbiased estimates exist and these estimates are displayed. Properties of the model and the estimates are discussed in both the complete and noncomplete density cases. Perhaps the most important contribution relates to the simplicity with which the theoretical methods treat the general variance components situation.