Functional Penalized Ridge Regression with a Parametric Partitioning Framework for High-Dimensional Data

This study proposes a novel partition-based functional ridge regression framework to address key challenges in functional linear models, including multicollinearity, overfitting, and interpretability. The coefficient function vector \beta(s) is partitioned into relevant \beta_1(s) and nuisance \beta_2(s) components, enabling selective penalization that distinguishes structurally important effects from redundant or weak signals. The proposed partitioning strategy serves as the key mechanism for adaptive penalization in the functional ridge framework. This partitioning allows the model to adaptively control regularization strength across di erent components of the functional covariates which is an essential step in high-dimensional settings where irrelevant functional predictors can inflate variance and obscure interpretability. Our proposed estimators, the Functional Ridge Estimator (FRE), the Functional Ridge Full Model (FRFM), and the Functional Ridge Sub-Model (FRSM), apply tailored ridge penalties ( $\lambda_1$; $\lambda_2$; $\lambda_3$) to regulate smoothness and shrinkage. Under standard regularity conditions, the estimators are shown to be consistent, asymptotically normal, and e cient. Monte Carlo experiments con rm that FRSM yields the lowest Integrated mean squared error (IMSE) in small samples, while FRFM o ers greater stability for moderate and large samples. An empirical application to Canadian weather data further validates the framework, demonstrating reduced variance in ation and improved interpretability.