Suppose \(Y_1,\dots ,Y_n\) are observations from an AR(p) process with mean \(\mu \) , e.g., \(Y_t=\mu +\phi _1(Y_{t-1}-\mu )+\cdots +\phi _p(Y_{t-p}-\mu )+Z_t\) , where \(\{Z_t\}\) is an IID sequence with mean zero, variance \(\sigma ^2\) , and common distribution function \(F_0(z)\) . Indexing by quantile \(q=F_0(z)\) , it will be shown that if the parameters are estimated via maximum likelihood (MLE) using only the first half of the observations ( \(Y_1,\dots ,Y_{\lfloor n/2\rfloor }\) ), then the empirical process based on all of the residuals will converge in distribution to B(q), where B is a standard Brownian bridge on [0, 1]. If all the data is used to estimate the parameters using MLE or least squares (LS), then the empirical process of the residuals will converge in distribution to B(q) plus a correction term. This is the content of [9] in the Gaussian case and of [8] in the more general case. Nevertheless, this correction term disappears when using the first half of the observations to estimate the parameters. These results may be viewed as extensions of the half-sample device of [6] and also connect with the results of [5] for the sample ACF.
