Title: Accuracy and validity of posterior distributions using the Cressie-Read empirical likelihoods,
Abstract: 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.