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Biometrika 2006 93(3):723-733; doi:10.1093/biomet/93.3.723
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© 2006 Biometrika Trust

Empirical-type likelihoods allowing posterior credible sets with frequentist validity: Higher-order asymptotics

Kai-Tai Fang1 and Rahul Mukerjee2

1 Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China. ktfang{at}math.hkbu.edu.hk, 2 Indian Institute of Management Calcutta, Joka, Diamond Harbour Road, Kolkata 700 104, India. rmuk1{at}hotmail.com

With reference to a general class of empirical-type likelihoods, we develop higher-order asymptotics for the frequentist coverage of Bayesian credible sets based on posterior quantiles and highest posterior density. These asymptotics, in turn, characterise members of the class that allow approximate frequentist validity of such sets. It is seen that the usual empirical likelihood does not enjoy this property up to the order of approximation considered here.

Key Words: Edgeworth expansion; Empirical likelihood; Highest posterior density region; Posterior quantile; Probability matching prior.

Received June 2005. Revised January 2006.


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