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.