In the context of scaling relations between Supermassive Black Holes and host-galaxy properties, we aim to enhance the comparison between \(M_{\bullet } - M_{G}\sigma ^{2}\) and \(M_{\bullet } - \sigma \) relations from a statistical point of view. First, it is suggested to take into account the predictive accuracy of the scaling relation, in addition to the classical measures of goodness of fit. Here, prediction accuracy is fairly evaluated according to a leave-one-out cross-validation strategy. Then, we spread more light on the analysis of residuals from the fitted scaling relation, in order to provide more useful information on the role played by the different variables in their correlation with the black hole mass. The findings from six samples are discussed.

在超大质量黑洞和宿主星系属性之间的缩放关系的背景下，我们旨在加强\(M_{\bullet } - M_{G}\sigma ^{2}\)和\(M_{\bullet } - \sigma \)从统计的角度来看关系。首先，除了经典的拟合优度度量之外，建议考虑缩放关系的预测准确性。在这里，根据留一法交叉验证策略公平地评估预测准确性。然后，我们从拟合的比例关系中传播更多关于残差分析的信息，以便提供有关不同变量在与黑洞质量的相关性中所起作用的更多有用信息。讨论了来自六个样本的发现。