Accuracy and validity of posterior distributions using the Cressie-Read empirical likelihoods

The class of Cressie-Read empirical likelihoods are constructed with weights derived at a minimum distance from the empirical distribution in the Cressie-Read family of divergences indexed by $\gamma$ under the constraint of an unbiased set of $M$-estimating equations. At first order, they provide valid posterior probability statements for any given prior, but the bias in coverage of the resulting empirical quantile is inversely proportional to the asymptotic efficiency of the corresponding $M$-estimator. The Cressie-Read empirical likelihoods based on the maximum likelihood estimating equations bring about quantiles covering with $O(n^{-1})$ accuracy at the underlying posterior distribution. The choice of $\gamma$ has an impact on the variance in small samples of the posterior quantile function. Examples are given for the $M$-type estimating equations of location and for the quasi-likelihood functions in the generalized linear models.