On the quantile process based on the autoregressive residuals

✍ Scribed by Sangyeol Lee

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 437 KB
• Volume: 67
• Category: Article
• ISSN: 0378-3758
• DOI: 10.1016/s0378-3758(97)00070-0

✦ Synopsis

Let {Xt} be the stationary AR(p) process satisfying the difference equation Xt = ~lXt-i +... + ~pXt-p + ~t, where {~t} is a sequence of lid random variables with mean zero and finite variance. Motivated by a goodness of fit test on the true errors {~t}, we are led to study the asymptotic behavior of the quantile process based on residuals (the residual quantile process). Particularly, we concentrate on the deviations between the residual quantile process and the empirical process based on the true errors. In this asymptotic study, it is shown that the deviations converge to zero in probability uniformly over certain intervals with specific order as sample size increases.

Here, these intervals are allowed to vary with the sample size n and converge to the unit interval as n goes to infinity. Then, based on our result and the strong approximation result of Csrrg6 and R~vrsz (1978), we propose a goodness of fit test statistic of which limiting distribution is the same as of a functional form of a standard Brownian bridge.