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We propose kernel estimator for the distribution function of unobservederrors in autoregressive time series, based on residuals computed by estimatingthe autoregressive coefficients with the Yule-Walker method. Under mildassumptions, we establish oracle efficiency of the proposed estimator, that is,it is asymptotically as efficient as the kernel estimator of the distributionfunction based on the unobserved error sequence itself. Applying the result ofWang, Cheng and Yang [J. Nonparametr. Stat. 25 (2013) 395-407], the proposedestimator is also asymptotically indistinguishable from the empiricaldistribution function based on the unobserved errors. A smooth simultaneousconfidence band (SCB) is then constructed based on the proposed smoothdistribution estimator and Kolmogorov distribution. Simulation examples supportthe asymptotic theory.

Oracally efficient estimation of autoregressive error distribution with simultaneous confidence band