A note on invariant quadratics

Invariant quadratics play an important role in inferences about variance components in linear models. Sometimes called “translation invariance” or “location invariance”, this property is widely defined in a way that assumes that the parameter set and the random variable in question fill the linear subspaces in which they take on values. In this note we show that such assumptions are unnecessary. We show that the defining characteristic of mean-invariant quadratic forms leads to a characterization of invariant quadratics that is less restrictive than the customary definition.