Oracally efficient estimation of autoregressive error distribution with simultaneous confidence band

TL;DR

This paper introduces a kernel estimator for the error distribution in autoregressive models that achieves oracle efficiency and includes a method for constructing a smooth simultaneous confidence band, validated by simulations.

Contribution

It develops an oracle-efficient kernel estimator for autoregressive error distribution using residuals and constructs a smooth confidence band, advancing nonparametric inference in time series.

Findings

Estimator is asymptotically as efficient as the ideal kernel estimator.

Proposed confidence band accurately covers the true distribution in simulations.

Method is supported by asymptotic theory and simulation results.

Abstract

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