A General Theorem on Decision Theory for Nonnegative Functionals: with Applications

Abstract

A distribution-free inadmissibility theorem is proved for estimating under quadratic loss nonnegative functionals that are allowed to depend on the unknown c.d.f. as well as the data. It follows from the theorem that, subject to finiteness of the risk, some natural estimators of the eigenvalues of many commonly occurring matrices in multivariate problems are inadmissible, typically from dimension 2, when samples are drawn from any multivariate elliptically symmetric distribution. As an example, if $X_1, \ldots, X_{k + 1}$ are i.i.d. observations from such a distribution with dispersion matrix $\Sigma,$ then the eigenvalues of $S/k$ are inadmissible for the eigenvalues of $\Sigma$ if a certain conditions holds where $S$ if the sample sum of squares and products matrix. It also follows from the theorem that in the general scale-parameter family, the best equivariant estimator of the scale-parameters is inadmissible for $p \geq 2,$ and some natural estimators of the losses of the best equivariant estimators are inadmissible, usually for $p \geq 2.$ The theorem also has certain applications to the Ferguson family of distributions, the multivariate $F$ distribution and for unbiased estimators in some families of distributions.