Sankhya: The Indian Journal of Statistics

1997, Volume 59, Series A, Pt. 2, 215-231

REFERENCE PRIORS UNDER THE CHI-SQUARED DISTANCE

By

BERTRAND CLARKE, University of British Columbia, Vancouver

And

DONGCHU SUN, University of Missouri-Columbia, Columbia

SUMMARY. For smooth parametric families in exponential form equipped with a smooth prior density on a real parameter q , reference priors for use with independent, identically distributed data are obtained by maximising the expected Chi-squared distance between a prior density and its corresponding posterior density. We identify an asymptotic expansion for this Chi-squared distance and use it to define a functional which can be optimised so as to obtain a reference prior. Performing this optimization for a unidimensional parameter leads to a prior which is proportional 1/Ö detI(q ), where q is a d-dimensional real parameter and I(q ) is the Fisher information matrix. We argue therelevance of the Chi-squared distance by noting its relationship with the Chi-squared goodness-of-fit statistic and present computational results to show how this Chi-squared reference prior performs for d = 1. In addition, we briefly consider the case where nuisance parameters are present. Finally, we discuss the importance of the choice of measure of distance on prior selection and evaluation.

 

AMS (1991) subject classification. 62A15, 62F15, 62C10.

Key words and phrases. Reference priors, distance measures, prior selection.

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