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Uniform in Bandwidth Consistency of the L1-Modal Regression Estimator for High-Dimensional Data
Fatimah A Almulhim1, Mohammed B Alamari2, Ali Laksaci2
1Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia.
Entropy (Basel, Switzerland)
|May 26, 2026
Summary
This study introduces a robust nonparametric estimator for conditional mode in functional regression using L1-quantile regression. The method offers improved stability and consistency for functional data analysis, aiding automatic bandwidth selection.
Area of Science:
- Statistics
- Functional Data Analysis
Background:
- Classical density-based methods for conditional mode estimation can be sensitive to outliers and heavy-tailed distributions.
- Functional regression involves covariates that are functions, posing unique challenges for estimation.
Purpose of the Study:
- To develop and analyze a novel nonparametric estimator for the conditional mode in functional regression.
- To establish strong consistency and convergence rates for the proposed kernel estimator, uniformly over bandwidth parameters.
Main Methods:
- Utilizing a quantile regression approach with an L1-structure for enhanced robustness.
- Employing kernel estimation techniques within the functional regression framework.
- Developing theoretical guarantees under general regularity conditions on functional regressors and model components.
Main Results:
- The proposed L1-quantile regression estimator demonstrates improved robustness against outliers and heavy-tailed errors.
- Strong consistency and explicit convergence rates are established for the kernel estimators, uniformly in bandwidth.
- The theoretical findings support automatic bandwidth selection procedures in high-dimensional functional data.
Conclusions:
- The new estimator provides a stable and reliable method for conditional mode estimation in complex functional data settings.
- This research offers theoretical backing for practical data-driven bandwidth selection in functional data analysis.
- The study validates the approach through simulations and real-world data applications.
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