Lower bounds on Bayes risks for estimating a normal variance: With applications†

TLDR: Brown and Gajek (1990) gave useful lower bounds on Bayes risks, which improved on earlier bounds by various authors as mentioned in this paper, such as the information inequality, using convexity of appropriate functionals instead of information inequality.

Abstract: Brown and Gajek (1990) gave useful lower bounds on Bayes risks, which improve on earlier bounds by various authors. Many of these use the information inequality. For estimating a normal variance using the invariant quadratic loss and any arbitrary prior on the reciprocal of the variance that is a mixture of Gamma distributions, we obtain lower bounds on Bayes risks that are different from Borovkov-Sakhanienko bounds. The main tool is convexity of appropriate functionals as opposed to the information inequality. The bounds are then applied to many specific examples, including the multi-Bayesian setup (Zidek and his coauthors). Subsequent use of moment theory and geometry gives a number of new results on efficiency of estimates which are linear in the sufficient statistic. These results complement earlier results of Donoho, Liu and MacGibbon (1990), Johnstone and MacGibbon (1992) and Vidakovic and DasGupta (1994) for the location case. Brown et Gajek (1990) ont donne des limites inferieures utiles aux risques de Bayes, qui ameliorent les limites donnees precedemment par differents auteurs. Plusieurs d'entre eux utilisent l'inegalite d'information. En estimant une variance normale a l'aide d'une fonction de perte quadratique invariante et d'un a priori arbitraire sur l'inverse de la variance, qui est un melange de distributions Gamma, on obtient des limites inferieures sur les risques de Bayes qui sont differentes des limites de Borovkov-Sakhanienko. L'outil principal est la convexite de fonctionnels appropries, et non l'inegalite de l'information. Les limites sont alors appliquees a differents exemples, dont a la situation multi-Bayesienne (Zidek et co-auteurs). L'utilisation subsequente de la theorie des moments et de la geometrie donne un certain nombre de nouveaux resultats sur l'efficacite des estimations, qui sont lineaires dans la statistique exhaustive. Ces resultats completent les resultats obtenus precedemment par Donoho, Liu et MacGibbon (1990), Johnson et MacGibbon (1992) et Vidakovic et DasGupta (1994) pour le cas de la position.