Asymptotic of the L_r-norm of density estimators in the autoregressive time series

Abstract

In this article, we consider autoregressive models with independent identically distributed innovations. We compare the kernel density estimator of fitted residuals with the theoretical kernel density estimator based on unobserved innovations. We show that the L _r -norm of the difference is asymptotically negligible.

Keywords:

• L _r -norms
• AR(p)
• stationarity
• converges in probability

AMS Mathematics Subject Classification (2000) :

• 62G07
• 60F05

Acknowledgements

This research was supported by the National Natural Science Foundation of China (No. 10771192 & 10671176) (National Natural Science Foundation of China Grant no. 11026087 and the Humanities and Social Sciences Foundation of the Ministry of Education of China Grant no. 10YJC910010, as of 1 January 2011).