Nonparametric estimation of coefficient function derivatives in varying coefficient models

Junfeng Huo; Mingquan Wang; Xiuqing Zhou; Junfeng Huo; Mingquan Wang; Xiuqing Zhou

doi:10.3934/math.2025526

AIMS Mathematics

2025, Volume 10, Issue 5: 11592-11626. doi: 10.3934/math.2025526

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Research article

Nonparametric estimation of coefficient function derivatives in varying coefficient models

1.
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
2.
School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

Received: 21 March 2025 Revised: 06 May 2025 Accepted: 12 May 2025 Published: 21 May 2025
MSC : 62F12, 62G05, 62H12

In varying-coefficient models, accurately estimating the derivatives of coefficient functions is crucial, particularly for optimal bandwidth selection and confidence interval construction. Despite its importance, methods have largely ignored derivative estimation. In this paper, we addresse this gap by proposing a novel weighted difference quotient approach to estimate both first and second order derivatives of coefficient functions under the differences in the smoothness of the coefficient functions. We derived the asymptotic properties of our estimator and introduced a data-driven tuning parameter selection method. Simulations and real-data analyses confirmed the superior performance of our approach compared to existing techniques. Our method provides the most accurate estimates, effectively estimating both the first and second derivatives.
- varying coefficient model,
- derivative estimation,
- asymptotic properties,
- symmetric quotients method,
- kernel method
Citation: Junfeng Huo, Mingquan Wang, Xiuqing Zhou. Nonparametric estimation of coefficient function derivatives in varying coefficient models[J]. AIMS Mathematics, 2025, 10(5): 11592-11626. doi: 10.3934/math.2025526

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Abstract

In varying-coefficient models, accurately estimating the derivatives of coefficient functions is crucial, particularly for optimal bandwidth selection and confidence interval construction. Despite its importance, methods have largely ignored derivative estimation. In this paper, we addresse this gap by proposing a novel weighted difference quotient approach to estimate both first and second order derivatives of coefficient functions under the differences in the smoothness of the coefficient functions. We derived the asymptotic properties of our estimator and introduced a data-driven tuning parameter selection method. Simulations and real-data analyses confirmed the superior performance of our approach compared to existing techniques. Our method provides the most accurate estimates, effectively estimating both the first and second derivatives.

References

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