The L_p consistency of error density estimator in censored linear regression

ABSTRACT

This paper considers, under L_p-norm, the global property for the smooth estimator of error density function in linear regression with right censored data. For any 1 ⩽ p < ∞, we investigate the L_p-norm of the difference between the kernel density estimator ${\widehat{f}}_n\left(t\right)$ and the true density function f(t). We obtain the convergence rate (in probability) for ${\int }_0^T{|{\widehat{f}}_n\left(t\right)-f\left(t\right)|}^{p}d\mu \left(t\right),1\le p<\infty ,0\le T<\infty$, where μ is a measure on the Borel sets of the real line.

KEYWORDS:

• Censored linear regression
• Kaplan-Meier estimator
• Kernel density estimation
• L_p-norm
• Right censored data

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 60F25
• Secondary: 62G07

Acknowledgments

I am grateful to the Editor in Chief and the reviewer for their helpful comments and suggestions which greatly improved the presentation of this article.