Sankhya: The Indian Journal of Statistics

2007, Volume 69, Pt. 4, 764--779

Kernel Density Estimation and Extended CLT and SLLN in ARCH(p)-Time Series

Fuxia Cheng, Illinois State University, Normal, USA

SUMMARY. In this paper we consider the estimation of the innovation density and the asymptotics of the sum of residuals and the sum of squared residuals in ARCH(p)-time series. We obtain the weak and strong uniform consistency of the kernel density estimators based on the residuals. We extend the Central Limit Theorem (CLT) and the Strong Law of Large Number (SLLN) to the average of residuals. For the average of squared residuals, we show its weak and strong consistency to the innovation variance.

AMS (2000) subject classification. Primary 62M10, 62M09; seconday 62F05, 62F15.

Key words and phrases. ARCH(p)-time series, Kernel density estimation, residuals, CLT, SLLN.

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