Asymptotic properties of conditional U-statistics using delta sequences

Abstract

Stute (1991) introduced a class of so-called conditional U-statistics, which may be viewed as a generalization of the Nadaraya-Watson estimates of a regression function. Stute proved their strong pointwise consistency to: $$r^{(k)}(\phi ,\mathbf{t}):=\mathbb{E}[\phi (Y_1,\dots ,Y_k)|(X_1,\dots ,X_k)=\mathbf{t}], \text{for}\mathrm{ }\mathbf{t}\in {\mathbb{R}}^{dk}.$$ This article deals with a quite general non parametric statistical curve estimation setting including the Stute estimator as a particular case. The class of “delta sequence estimators” is defined and treated here. This class includes also the orthogonal series and histogram methods. The theoretical results concerning the exponential inequalities and the asymptotic normality, established in this article, are (or will be) key tools for many further developments in functional estimation. As a by-product of our proofs, we state consistency results for the delta sequences conditional U-statistics estimator, under the random censoring. Potential applications include discrimination problems, metric learning and multipartite ranking, Kendall rank correlation coefficient, generalized U-statistics, and set indexed conditional U-statistics.

Keywords:

• Non parametric estimation
• U-statistics
• conditional distribution
• delta sequences
• kernel estimation
• machine learning problems

2020 Mathematics Subject Classification:

• 60F05
• 60G15
• 60G10
• 62G05
• 62G07
• 62H12

Declarations

Conflict of interest

The authors declare that they have no conflict of interest.

CRediT author statement

Amel NEZZAL: Conceptualization, Methodology, Investigation, Writing - Original Draft, Writing - Review & Editing. Salim BOUZEBDA: Conceptualization, Methodology, Investigation, Writing - Original Draft, Writing - Review & Editing. Both authors contributed equally to this work.

Acknowledgments

The authors would like to thank the Editor-in-Chief, Associate Editor and the three referees for their constructive comments that lead to numerous improvements over a previous version. The second author gratefully acknowledges the funding received towards his PhD from the Algerian government PhD fellowship.