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来拟合线性回归模型，其拟合值对设计矩阵的可逆线性变换具有不变性。这一不变性在实践中尤为重要，例如当预测变量包含分类变量及其交互项时。该方法与岭惩罚最小二乘法具有相同的计算成本，但后者缺乏这种不变性。我们推导了总体拟合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。数值实验与数据实例表明，我们的方法与岭惩罚最小二乘法表现相近。