Estimation and diagnostic for single-index partially functional linear regression model with $p$-order autoregressive skew-normal errors

This paper introduced a novel single-index partially functional linear regression model with $p$-order autoregressive skew-normal errors, addressing the dual challenges of autocorrelation and skewness in high-dimensional functional data. We proposed an innovative EM-CALS algorithm, which synergizes the expectation-maximization (EM) framework with conditional adaptive least squares to estimate constrained parameters and functional components efficiently. The algorithm uniquely addressed the unit-norm constraint of single-index vectors through reparameterization, thereby overcoming the limitations of conventional EM in nonlinear optimization. Comprehensive simulations demonstrated EM-CALS's superiority over the two-step iterative least squares (TSILS) algorithm, achieving reductions of 1.19% in root mean square error (RMSE), 2.45% in mean square error (MSE), 38.00% in root average squared errors ( ${\text{RASE}}_1$), and 51.35% in ${\text{RASE}}_2$, respectively, demonstrating a clear advantage in enhancing prediction accuracy. Additionally, we conducted residual analysis based on conditional quantiles, considering the skew-normal distribution and autocorrelation of the residuals, and performed local influence analysis using the Q-function in the EM algorithm. The efficiency of the EM-CALS algorithm was validated through simulation studies. Finally, the methodology was applied to an empirical dataset from photovoltaic power forecasting, underscoring its practical applicability. This work bridged significant gaps in functional data analysis by simultaneously addressing dimension reduction, temporal dependence, and distributional asymmetry, which are crucial challenges in modern energy analytics and biomedical studies.