Higher-order asymptotics of minimax estimators for time series

We consider the minimax estimation of time series in view of higher-order asymptotic theory. Under the framework of Bayesian inference, we focus on the Bayes estimator and the Bayesian Whittle estimator for parameter estimation. It is shown that these estimators are minimax with respect to the Bayes risk of higher-order bias appeared in their asymptotic expansion. The minimax problem in the boundary issue with parameter on the boundary of parameter space is also discussed. Our theoretical discovery is justified by simulation studies even when the sample size is small.