On the existence of unbiased nonnegative estimates of variance convariance components

The existence of unbiased nonnegative definite quadratic estimates for linear combinations of variance covariance components is characterized by means of the natural parameter set in a residual model. In the presence of a quadratic subspace condition the following disjunction for nonnegative estimability is derived: either standard methods suffice, or the concepts of unbiasedness and nonnegative definiteness are incompatible. For the case of a single variance component it is shown that unbiasedness and nonnegative definiteness always entail a reduction to a trivial model in which the variance component under investigation is the sole remaining parameter. Several examples illustrate these results.