This study develops a functional Liu-type shrinkage estimator (fLiu) for scalar-on-function regression in the presence of strong multicollinearity and high-dimensional functional predictors. The approach extends the classical Liu estimator to the functional setting by combining directional shrinkage with smoothness regularization, providing flexible control over the bias-variance trade-off. Theoretical analysis is used to examine the behavior of the estimator and the associated parameter selection problem. In particular, an explicit mean squared error (MSE) decomposition is derived, characterizing the risk of the estimator in terms of variance reduction and shrinkage bias. This further yields an explicit optimal choice of the shrinkage parameter of the fLiu estimator through a one-dimensional convex risk minimization problem, leading to a practical plug-in tuning rule. Moreover, it is shown that in high-dimensional (underdetermined) settings, commonly used criterion such as GCV (and equivalently PRESS/LOO-CV) become constant with respect to the parameter d, thus uninformative for tuning. This provides a theoretical explanation for the predominant focus on the overdetermined regime in existing Liu-type methods. Numerical results demonstrate that the estimator achieves competitive predictive accuracy relative to existing methods. Implementation is carried out in R using the fda package, and in Python via the fLiu.py package developed for this study.

翻译：本研究提出了一种针对强多重共线性与高维函数型预测变量的标量-函数回归中的函数型刘收缩估计量（fLiu）。该方法通过将方向性收缩与平滑正则化相结合，将经典刘估计量拓展至函数型设定，实现了对偏差-方差权衡的灵活控制。通过理论分析考察了该估计量的行为及相关参数选择问题。特别地，推导出均方误差（MSE）的显式分解，从方差缩减与收缩偏差两个维度刻画了估计量的风险特征。进一步通过一维凸风险最小化问题得到fLiu估计量收缩参数的显式最优选择，从而形成实用的插件式调参准则。此外，理论证明在高维（欠定）设定下，常用准则（如GCV及等价的PRESS/LOO-CV）关于参数d为常数，因此无法用于调参——这为现有刘型方法主要聚焦于过定场景提供了理论解释。数值实验表明，该估计量相较于现有方法具有竞争力的预测精度。实现基于R的fda包以及本研究开发的Python包fLiu.py。