Additive regression model for stationary and ergodic continuous time processes

Abstract

The main purpose of the present work is to introduce and investigate a simple kernel procedure based on marginal integration that estimates the regression function for stationary and ergodic continuous time processes in the setting of the additive model introduced by Stone (1985). We obtain the uniform almost sure consistency with exact rate and the asymptotic normality of the kernel-type estimators of the components of the additive model. Asymptotic properties of these estimators are obtained, under mild conditions, by means of martingale approaches. Finally, a general notion of the bootstrapped additive components, constructed by exchangeably weighting sample, is presented.

Keywords:

• Additive model
• consistency
• ergodicity
• Kernel-type estimators
• marginal integration
• martingale differences
• normality
• regression function
• stationarity.

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 60A10
• 60G09
• 62G08
• 62G07
• Secondary 62H05
• 62J02

Acknowledgments

The authors are grateful to the Editor, an associate editor, and the referees for several insightful suggestions that led to a significant improvement of the presentation and the correction of a technical argument in the original proof.