Comparing UMVU and ML estimators of the generalized variance for natural exponential families

Abstract

In this paper we compare the uniformly minimum variance unbiased (UMVU) estimator and maximum likelihood (ML) estimator of the generalized variance in the context of natural exponential families (NEFs) on , d>1. We conjecture that for irreducible NEFs the proportionality holds if and only if the generalized variance has a specific form. In particular, we show that the estimators are proportional in the simple and homogeneous quadratic NEFs and prove that the UMVU estimator is preferable in terms of mean squared error except for the case of multinomial family.

Keywords:

• Generalized variance
• Inverse Gaussian distribution
• Wishart distribution

Acknowledgements

We thank the Editor and the anonymous referees for their valuable comments. We dedicate special thanks to Gérard Letac for his useful conversations. Part of this work was done while the first author was at Ecole Nationale de Statistique et de l'Analyse de l'Information as a Visiting Scientist, with the support of CREST.