A hybrid particle swarm optimization framework for robust estimation in partially linear models

Partially linear models (PLMs) provide a flexible framework for modeling complex relationships by combining parametric and nonparametric components. However, traditional estimation procedures are often based on multi-step algorithms that may suffer from convergence issues and sensitivity to data contamination. To overcome these limitations, this study proposes a hybrid Particle Swarm Optimization (PSO) algorithm for the simultaneous estimation of the parameters of the PLM with normal errors (PLMN). The proposed approach is based on Inertia Weight PSO (IWPSO) within a penalized likelihood framework, together with Huber-type adaptive weighting, BIC-based smoothing parameter selection, and SQP-based local refinement. The performance of the proposed method is evaluated through extensive Monte Carlo simulations based on Doppler and Sine functions under both standard normal and contaminated normal error distributions, including a component-wise comparison to assess the individual contributions of the proposed framework. The results indicate that the proposed method performs comparably to PLMN under standard conditions while providing substantially improved robustness under contaminated errors, particularly in estimating the nonparametric component. The component-wise analysis further shows that these improvements result from the combined effect of adaptive Huber weighting and the proposed optimization strategy. Finally, the practical applicability of the proposed method is demonstrated using a real-world ragweed pollen concentration dataset. The empirical results, supported by cross-validation, indicate that the proposed Hybrid PSO provides a stable and robust alternative to conventional PLMN estimation.