Volume 23, 2019
|Page(s)||464 - 491|
|Published online||07 August 2019|
On the Bickel–Rosenblatt test of goodness-of-fit for the residuals of autoregressive processes
Institut de Mathématiques de Toulouse, UMR 5219, CNRS UT2J, Université de Toulouse,
2 VNUHCM - University of Science, Ho Chi Minh City, Viet Nam.
3 Laboratoire angevin de recherche en mathématiques, LAREMA, UMR 6093, CNRS, UNIV Angers, SFR MathSTIC, 2 Bd Lavoisier, 49045 Angers Cedex 01, France.
* Corresponding author: firstname.lastname@example.org
Accepted: 14 September 2018
We investigate in this paper a Bickel–Rosenblatt test of goodness-of-fit for the density of the noise in an autoregressive model. Since the seminal work of Bickel and Rosenblatt, it is well-known that the integrated squared error of the Parzen–Rosenblatt density estimator, once correctly renormalized, is asymptotically Gaussian for independent and identically distributed (i.i.d.) sequences. We show that the result still holds when the statistic is built from the residuals of general stable and explosive autoregressive processes. In the univariate unstable case, we prove that the result holds when the unit root is located at − 1 whereas we give further results when the unit root is located at 1. In particular, we establish that except for some particular asymmetric kernels leading to a non-Gaussian limiting distribution and a slower convergence, the statistic has the same order of magnitude. We also study some common unstable cases, like the integrated seasonal process. Finally, we build a goodness-of-fit Bickel–Rosenblatt test for the true density of the noise together with its empirical properties on the basis of a simulation study.
Mathematics Subject Classification: 62M10 / 62F03 / 62F12 / 62G08
Key words: Autoregressive process / Bickel–Rosenblatt statistic / goodness-of-fit / hypothesis testing / nonparametric estimation / Parzen–Rosenblatt density estimator / residual process
© The authors. Published by EDP Sciences, SMAI 2019
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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