Quadratic unbiased estimation of variance components for the one-way classification

Abstract

SUMMARY The paper deals with the quadratic estimation of the components of variance associated with the one-way random classification where the effects are taken to be independently and normally distributed and where the class numbers are unequal. An estimator is said to be inquadmissible or quadmissible depending on whether or not there exists a second quadratic estimator having the same expectation and smaller variance. A quadratic estimator is shown to be quadmissible only if it is a function of the minimal sufficient statistic of a certain prescribed form. Certain invariance criteria are introduced. Equations are given for determining locally best quadratic unbiased estimators. Conditions are provided which aid in ascertaining the quadmissibility or inquadmissibility of any given invariant quadratic unbiased estimator of the upper variance component. The theory for quadratic unbiased estimation of the variance components in ordinary random effects models having equal subclass numbers is essentially complete. It is known that the analysis-of-variance estimators of the components in such models are minimumvariance quadratic unbiased. If, in addition, the random effects are taken to be normally distributed, then these estimators are minimum-variance unbiased (Graybill, 1961). In contrast, the theory for those random-effects models having unequal subclass numbers is rather rudimentary. The traditional approach to the estimation of the variance components in such cases is to select several quadratic functions of the data in such a way that, when each function is set equal to its expectation, the resulting system of equations yields a unique solution for the components. Usually, the selected quadratics are among those associated with some particular type of analysis of variance. The only known optimality properties of the estimators constructed in this fashion are that they are quadratic functions of the data, are unbiased, and, in the special case of equal subclass numbers, simplify for the most part to the estimators that are 'best' for that special case. The present paper consists of a study of quadratic unbiased estimation of the variance components associated with the ordinary unbalanced one-way random classification where the random effects are taken to be normally distributed. Recall that one estimator is said