Estimation and diagnostic for a skewed generalized normal partially linear models

Xue Wang; Weihu Cheng; Clécio S. Ferreira; Xue Wang; Weihu Cheng; Clécio S. Ferreira

doi:10.3934/math.2025703

AIMS Mathematics

2025, Volume 10, Issue 7: 15698-15719. doi: 10.3934/math.2025703

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Research article

Estimation and diagnostic for a skewed generalized normal partially linear models

1.
School of Mathematics, Statistics and Mechanics, Beijing University of Technology, Beijing 100124, China
2.
College of Science, Qiqihar University, Qiqihar 161006, China
3.
Department of Statistics, Federal University of Juiz de Fora, Juiz de Fora, Brazil

Received: 08 April 2025 Revised: 26 June 2025 Accepted: 02 July 2025 Published: 09 July 2025
MSC : 62J02, 62J20

Partially linear models (PLMs) are widely employed in scientific research to analyze hybrid parametric-nonparametric relationships. However, their conventional reliance on symmetric error distributions severely limits applicability to real-world phenomena characterized by pronounced asymmetry and heavy-tailed behavior. To address this gap, we propose a novel PLM framework incorporating skewed generalized normal (SGN) distributed errors, which simultaneously accommodates extreme skewness and heavy-tailed attributes beyond the capabilities of symmetric or skew-normal (SN) specifications. Methodologically, we develop a penalized expectation-maximization (EM) algorithm with provable convergence guarantees and integrated adaptive smoothing selection, effectively resolving optimization instability in high-dimensional settings. Furthermore, we establish a unified diagnostic system that synergizes geometric leverage calculus with local influence analysis to systematically evaluate model robustness against perturbations and outliers. Extensive simulation studies demonstrate the framework's superior estimation accuracy compared to conventional symmetric and SN-based alternatives. Empirical validations based on real-world datasets reveal statistically significant improvements in model fit while maintaining interpretability.
- skewed generalized normal distribution,
- partially linear model,
- EM algorithm,
- local influence,
- generalized leverage
Citation: Xue Wang, Weihu Cheng, Clécio S. Ferreira. Estimation and diagnostic for a skewed generalized normal partially linear models[J]. AIMS Mathematics, 2025, 10(7): 15698-15719. doi: 10.3934/math.2025703

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Abstract

Partially linear models (PLMs) are widely employed in scientific research to analyze hybrid parametric-nonparametric relationships. However, their conventional reliance on symmetric error distributions severely limits applicability to real-world phenomena characterized by pronounced asymmetry and heavy-tailed behavior. To address this gap, we propose a novel PLM framework incorporating skewed generalized normal (SGN) distributed errors, which simultaneously accommodates extreme skewness and heavy-tailed attributes beyond the capabilities of symmetric or skew-normal (SN) specifications. Methodologically, we develop a penalized expectation-maximization (EM) algorithm with provable convergence guarantees and integrated adaptive smoothing selection, effectively resolving optimization instability in high-dimensional settings. Furthermore, we establish a unified diagnostic system that synergizes geometric leverage calculus with local influence analysis to systematically evaluate model robustness against perturbations and outliers. Extensive simulation studies demonstrate the framework's superior estimation accuracy compared to conventional symmetric and SN-based alternatives. Empirical validations based on real-world datasets reveal statistically significant improvements in model fit while maintaining interpretability.

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