We propose a penalized least-squares method to fit the linear regression model with fitted values that are invariant to invertible linear transformations of the design matrix. This invariance is important, for example, when practitioners have categorical predictors and interactions. Our method has the same computational cost as ridge-penalized least squares, which lacks this invariance. We derive the expected squared distance between the vector of population fitted values and its shrinkage estimator as well as the tuning parameter value that minimizes this expectation. In addition to using cross validation, we construct two estimators of this optimal tuning parameter value and study their asymptotic properties. Our numerical experiments and data examples show that our method performs similarly to ridge-penalized least-squares.

翻译：我们提出一种惩罚最小二乘法，用于拟合与设计矩阵可逆线性变换保持不变的线性回归模型的适配值。这种不变性在实践者涉及分类预测变量及其交互项时尤为重要。该方法在保持与岭惩罚最小二乘法相同计算量的前提下，克服了后者缺乏不变性的缺陷。我们推导了总体适配值向量与其收缩估计量之间的期望平方距离，以及使该期望最小化的调优参数值。除了使用交叉验证外，我们构建了该最优调优参数值的两种估计量，并研究了其渐近性质。数值实验和实例分析表明，所提方法与岭惩罚最小二乘法的性能相似。