Asymptotic results on the frequentist mean squared error of generalized Bayes point predictors

An asymptotic formula for the frequentist mean squared error (MSE) of generalized Bayes point predictors is worked out. This formula yields an explicit second-order admissibility result when the underlying parameter is scalar valued. We note that probability-matching priors, including Jeffreys' prior, may not always behave well with respect to the MSE of generalized Bayes point predictors. On the other hand, it is seen that priors that match the posterior and frequentist MSEs of such predictors can also keep the frequentist MSE small.