Oracally Efficient Estimation Of Innovation Quantile And Prediction Bounds For Autoregressive Time Series

An estimator is proposed for the quantile of autoregressive time series error distribution, based on kernel smoothing of Yule-Walker residuals. It is proved under mild assumptions that the quantile estimator is oracally e?cient as the infeasible sample quantile estimator based on unobserved errors, and thus follows the same asymptotic normal distribution. Prediction interval for future observation is constructed using the estimated quantiles and shown to possess asymptotically the prescribed con?dence level. Simulation examples support the asymptotic theory and an application to real data example is provided for illustration.