Estimation and diagnostic for a skewed generalized normal partially linear models

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.