On the Bickel–Rosenblatt test of goodness-of-fit for the residuals of autoregressive processes

¹ Institut de Mathématiques de Toulouse, UMR 5219, CNRS UT2J, Université de Toulouse, F-31058 Toulouse, France.
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: lagnoux@univ-tlse2.fr

Received: 19 December 2017
Accepted: 14 September 2018

Abstract

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.