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

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 indepently 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 quadratric 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.