Uniform convergence rate of the kernel regression estimator adaptive to intrinsic dimension in presence of censored data

ABSTRACT

The focus of the present paper is on the uniform in bandwidth consistency of kernel-type estimators of the regression function $\mathbb{E}\left(\mathrm{\Psi }\right(\mathbf{Y})\mid \mathbf{X}=\mathbf{x})$ derived by modern empirical process theory, under weaker conditions on the kernel than previously used in the literature. Our theorems allow data-driven local bandwidths for these statistics. We extend existing uniform bounds on kernel regression estimator and making it adaptive to the intrinsic dimension of the underlying distribution of $\mathbf{X}$ which will be characterising by the so-called intrinsic dimension. Moreover, we show, in the same context, the uniform in bandwidth consistency for nonparametric inverse probability of censoring weighted (I.P.C.W.) estimators of the regression function under random censorship. Statistical applications to the kernel-type estimators (density, regression, conditional distribution, derivative functions, entropy, mode and additive models) are given to motivate these results.

Keywords:

• Conditional empirical processes
• VC-classes
• Kernel-type estimators
• density function
• regression function
• censored data

Mathematics Subject Classifications:

• 60F05
• 60G15
• 60G10
• 62G08
• 62G07

Acknowledgments

The authors are indebted to the Editor-in-Chief, Associate Editor and the referees for their very valuable comments, suggestions careful reading of the article which led to a considerable improvement of the manuscript.

CRediT author statement

Thouria El-hadjali : Conceptualisation, Methodology, Investigation, Writing - Original Draft, Writing - Review & Editing.

Salim Bouzebda : Conceptualisation, Methodology, Investigation, Writing - Original Draft, Writing - Review & Editing.

Disclosure statement

No potential conflict of interest was reported by the authors.