Scalar-on-Function Relative Error Regression for Weak Dependent Case
Abstract
:1. Introduction
2. The Re-Regression Model and Its Estimation
3. The Consistency of the Kernel Estimator
- (D1)
- For all , and .
- (D2)
- For all ,
- (D3)
- The covariance coefficient is , such that
- (D4)
- K is the Lipschitzian kernel function, which has as support and satisfies the following:
- (D5)
- The endogenous variable Y gives:
- (D6)
- For all ,
- (D7)
- There exist and
- Brief comment on the conditions: Note that the required conditions stated above are standard in the context of Hilbertian time series analysis. Such conditions explore the fundamental axes of this contribution. The functional path of the data is explored through the condition (D1), the nonparametric nature of the model is characterized by (D2), and the correlation degree of the Hilbertian time series is explored by conditions (D3) and (D6). The principal parameters used in the estimator, namely the kernel and the bandwidth parameter, are explored through the conditions, (D4), (D5), and (D6). Such conditions are of a technical nature. They allow for retaining the usual convergence rate in nonparametric Hilbertian time series analysis.
- (K1)
- has a bounded derivative on ;
- (K2)
- The function , such that
- (K3)
- There exist and such that
4. Smoothing Parameter Selection
4.1. Leave-One-Out Cross-Validation Principle
4.2. Bootstrap Approach
- Step 1.
- We choose an arbitrary bandwidth (resp. ), and we calculate (resp. ).
- Step 2.
- We estimate (resp. ).
- Step 3.
- We create a sample of residual (resp. ) from the distribution
- Step 4.
- We reconstruct the sample (resp.
- Step 5.
- We use the sample to calculate and to calculate .
- Step 6.
- We repeat the previous steps times and put (resp. ), the estimators, at the replication r.
- Step 7.
- We select h (resp. k) according to the criteria
5. Computational Study
5.1. Empirical Analysis
5.2. A Real Data Application
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
- The first case is ; based on the definition of quasi-association, we obtain
- The second one is where . In this case, we have
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Months | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
Outliers | 15 | 26 | 13 | 5 | 24 | 25 | 7 | 9 | 11 | 8 | 9 | 15 |
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Chikr Elmezouar, Z.; Alshahrani, F.; Almanjahie, I.M.; Kaid, Z.; Laksaci, A.; Rachdi, M. Scalar-on-Function Relative Error Regression for Weak Dependent Case. Axioms 2023, 12, 613. https://doi.org/10.3390/axioms12070613
Chikr Elmezouar Z, Alshahrani F, Almanjahie IM, Kaid Z, Laksaci A, Rachdi M. Scalar-on-Function Relative Error Regression for Weak Dependent Case. Axioms. 2023; 12(7):613. https://doi.org/10.3390/axioms12070613
Chicago/Turabian StyleChikr Elmezouar, Zouaoui, Fatimah Alshahrani, Ibrahim M. Almanjahie, Zoulikha Kaid, Ali Laksaci, and Mustapha Rachdi. 2023. "Scalar-on-Function Relative Error Regression for Weak Dependent Case" Axioms 12, no. 7: 613. https://doi.org/10.3390/axioms12070613