Uniform Consistency for Functional Conditional U-Statistics Using Delta-Sequences
Abstract
:1. Introduction
2. Preliminaries and Estimation Procedure
- (C.1)
- For each γ in such a way that :
- (C.2)
- There exists a positive constant , in such a way that
- (C.3)
- There exist and , in such a way that
- (C.4)
- For any :
2.1. Examples of Delta Sequence
2.2. Conditions and Comments
- (C.5)
- We assume that for and
- (C.6)
- Suppose that and that
- (C.7)
- We assume the following usual boundedness condition:
- (C.7’)
- The function is unbounded and fullfils for some
- (C.8)
- For every :
- (C.9)
- The regression operator is Lipschitzian in the following sense: in such a way that, for any and , we have
2.3. Comments on the Assumptions
- (C.7’)
- We denote by a nonnegative continuous function, increasing on , and such that, for some , ultimately as ,For each , we define by . We assume further that:
- (i)
- for some ;
- (ii)
- for some .
3. Some Asymptotic Results
3.1. Uniform Consistency of Functional Conditional U-Statistics
3.2. Uniform Strong Consistency with Rates
4. Conditional -Statistics for Censored Data
- ()
- C and are independent.
5. Applications
5.1. Kendall Rank Correlation Coefficient
5.2. Discrimination Problems
5.3. Metric Learning
5.4. Time Series Prediction from Continuous Set of Past Values
5.5. Example of U-Kernels
6. Concluding Remarks
7. Mathematical Development
7.1. Truncated Part
7.2. Remainder Part
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
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Bouzebda, S.; Nezzal, A.; Zari, T. Uniform Consistency for Functional Conditional U-Statistics Using Delta-Sequences. Mathematics 2023, 11, 161. https://doi.org/10.3390/math11010161
Bouzebda S, Nezzal A, Zari T. Uniform Consistency for Functional Conditional U-Statistics Using Delta-Sequences. Mathematics. 2023; 11(1):161. https://doi.org/10.3390/math11010161
Chicago/Turabian StyleBouzebda, Salim, Amel Nezzal, and Tarek Zari. 2023. "Uniform Consistency for Functional Conditional U-Statistics Using Delta-Sequences" Mathematics 11, no. 1: 161. https://doi.org/10.3390/math11010161
APA StyleBouzebda, S., Nezzal, A., & Zari, T. (2023). Uniform Consistency for Functional Conditional U-Statistics Using Delta-Sequences. Mathematics, 11(1), 161. https://doi.org/10.3390/math11010161