Robust kernel regression function with uncertain scale parameter for high dimensional ergodic data using $ k $-nearest neighbor estimation

Fatimah Alshahrani; Wahiba Bouabsa; Ibrahim M. Almanjahie; Mohammed Kadi Attouch; Fatimah Alshahrani; Wahiba Bouabsa; Ibrahim M. Almanjahie; Mohammed Kadi Attouch

doi:10.3934/math.2023655

AIMS Mathematics

2023, Volume 8, Issue 6: 13000-13023. doi: 10.3934/math.2023655

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Research article

Robust kernel regression function with uncertain scale parameter for high dimensional ergodic data using $ k $-nearest neighbor estimation

1.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, Riyadh 11671, Saudi Arabia
2.
Laboratory of Statistics and Stochastic Processes, University of Djillali Liabes BP 89, Sidi Bel Abbes 22000, Algeria
3.
Department of Mathematics, College of Science, King Khalid University, Abha 62223, Saudi Arabia

Received: 09 January 2023 Revised: 17 February 2023 Accepted: 22 February 2023 Published: 03 April 2023
MSC : 62H12, 62G07, 62G35, 62G20

In this paper, we consider a new method dealing with the problem of estimating the scoring function $ \gamma_a $, with a constant $ a $, in functional space and an unknown scale parameter under a nonparametric robust regression model. Based on the $ k $ Nearest Neighbors ($ k $NN) method, the primary objective is to prove the asymptotic normality aspect in the case of a stationary ergodic process of this estimator. We begin by establishing the almost certain convergence of a conditional distribution estimator. Then, we derive the almost certain convergence (with rate) of the conditional median (scale parameter estimator) and the asymptotic normality of the robust regression function, even when the scale parameter is unknown. Finally, the simulation and real-world data results reveal the consistency and superiority of our theoretical analysis in which the performance of the $ k $NN estimator is comparable to that of the well-known kernel estimator, and it outperforms a nonparametric series (spline) estimator when there are irrelevant regressors.
- asymptotic normality,
- functional data analysis,
- robust regression estimation,
- $ k $-NN method,
- financial data,
- kernel method
Citation: Fatimah Alshahrani, Wahiba Bouabsa, Ibrahim M. Almanjahie, Mohammed Kadi Attouch. Robust kernel regression function with uncertain scale parameter for high dimensional ergodic data using $ k $-nearest neighbor estimation[J]. AIMS Mathematics, 2023, 8(6): 13000-13023. doi: 10.3934/math.2023655

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Abstract

In this paper, we consider a new method dealing with the problem of estimating the scoring function $ \gamma_a $, with a constant $ a $, in functional space and an unknown scale parameter under a nonparametric robust regression model. Based on the $ k $ Nearest Neighbors ($ k $NN) method, the primary objective is to prove the asymptotic normality aspect in the case of a stationary ergodic process of this estimator. We begin by establishing the almost certain convergence of a conditional distribution estimator. Then, we derive the almost certain convergence (with rate) of the conditional median (scale parameter estimator) and the asymptotic normality of the robust regression function, even when the scale parameter is unknown. Finally, the simulation and real-world data results reveal the consistency and superiority of our theoretical analysis in which the performance of the $ k $NN estimator is comparable to that of the well-known kernel estimator, and it outperforms a nonparametric series (spline) estimator when there are irrelevant regressors.

References

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