In case of multicollinearity, PRESS statistics has been proposed to be used in the selection of the ridge biasing parameter of the ridge estimator which is introduced as an alternative to BLUE. This newly proposed PRESS statistic for the ridge estimator, \(\textit{CPRESS}_{k}\), depends on the conditional ridge residual and can be computed once at a time by fitting the linear mixed model with all the observations. We also define \(R^2_{RidPred}\) statistic to evaluate the predictive ability of the ridge fit. Since the PRESS statistic for the BLUE is a special \(\textit{CPRESS}_{k}\) statistic, we indirectly also give closed form solution of the PRESS statistic for the BLUE. Then, we compared the predictive performance of the linear mixed model via the statistics \( \textit{CPRESS}_{k}\), \(GCV_{k}\) and \(C_{p}\) by considering a real data analysis and a simulation study where the optimal ridge biaisng parameter is obtained by minimizing each statistic. The study shows that the ridge predictors improve the predictive performance of a linear mixed model over BLUE in the presence of multicollinearity and each statistic gives a different optimum ridge biasing value and they show the best predictive performance at their optimum ridge biasing values. In addition, the simulation study has shown that the intensity of variance and multicollinearity is effective in determining the optimum ridge biasing value and this optimum ridge biasing value is effective on the superiority of the predictive performance of ridge estimator over BLUE.

在多重共线性的情况下，已建议使用 PRESS 统计来选择岭估计器的岭偏置参数，该岭估计器是作为 BLUE 的替代方案引入的。这个新提出的岭估计量的 PRESS 统计量\(\textit{CPRESS}_{k}\)取决于条件岭残差，并且可以通过将线性混合模型与所有观测值拟合来一次计算一次。我们还定义\(R^2_{RidPred}\)统计量来评估岭拟合的预测能力。由于蓝色的 PRESS 统计量是一个特殊的\(\textit{CPRESS}_{k}\)统计量，我们还间接给出了蓝色的 PRESS 统计量的封闭形式解。然后，我们通过统计数据\( \textit{CPRESS}_{k}\)、\(GCV_{k}\)和\(C_{p}\)比较线性混合模型的预测性能通过考虑实际数据分析和模拟研究，其中通过最小化每个统计量来获得最佳脊偏置参数。研究表明，在存在多重共线性的情况下，岭预测器比 BLUE 提高了线性混合模型的预测性能，并且每个统计量都给出了不同的最佳岭偏差值，并且它们在最佳岭偏差值下表现出最佳预测性能。此外，模拟研究表明，方差和多重共线性的强度可以有效地确定最佳岭偏置值，并且该最佳岭偏置值可以有效地提高岭估计器相对于BLUE的预测性能的优越性。