Simulating the behavior of a kernel M-estimator for left-truncated and associated model

Abstract

This paper investigates theoretical and numerical asymptotic properties of a nonparametric M-estimator’s regression function when the data are left truncated and associated. Using specific tools to this weak dependence, we study the strong uniform consistency of the proposed estimator and give its rate of convergence. A large Monte Carlo simulation study is carried out to comfort the good behavior of the M-estimator for different sample sizes and different truncation rates: First we show that the proposed estimator performs better than the Nadaraya-Watson estimator by looking at their respective closeness to the true univariate and bivariate regression function. Thereafter and to highlight the robustness of our estimator, we consider a regression model with a heavy-tailed error distribution and we compare the global mean square error of the two estimators. Finally, we study the behavior of the two estimators in the presence of outliers using the influence function to show the superiority of the M-estimator.

Keywords:

• Associated data
• Influence function
• M-estimator
• truncated data
• Uniform strong consistency

MATHEMATICS SUBJECT CLASSIFICATION:

• 62G05
• 62G20

Acknowledgements

The authors are indebted to an anonymous referee whose careful reading and constructive comments and suggestions helped to improve an earlier version of this paper.