On the nonexistence of a global nonnegative minimum bias invariant quadratic estimator of variance components

Abstract

We consider nonnegative invariant quadratic estimation of variance components and prove that if there does not exist a nonnegative quadratic unbiased estimator of variance components, there does not exist a nonnegative invariant quadratic estimator with a global minimum bias. The result demonstrates that any nonnegative invariant quadratic estimator can only achieve local minimum bias and explains existing numerical results.