Asymptotics of $L_\lambda$-Norms of ARCH(p) Innovation Density Estimators

Abstract

In this paper we consider, under L$_{\lambda}$-norm, the global property for a residual-based kernel estimator of the innovation density estimator in ARCH time series. For any $1 \leq \lambda <\infty$, we investigate the L$_{\lambda}$-norm of the difference between this estimator and a innovation-based kernel density estimator. For $\lambda>1$, this quantity is shown to be small enough so that the asymptotic distribution for the $L_\lambda$-norm of the difference between the innovation-based kernel estimator and the innovation density (which is known from the i.i.d. case) carries over to the difference between the residual-based kernel estimator and the innovation density. For $\lambda=1$, this is no longer the case as the above quantity is then of the same magnitude as the L ₁-norn of the difference between the innovation-based kernel estimator and the innovation density.