A note on the interpretation of the statistical analysis of the $M_{\bullet}-M_{G}\sigma ^{2}$ scaling relation

Abstract

In the context of scaling relations between Supermassive Black Holes and host-galaxy properties, we aim to enhance the comparison between M • −M G σ 2 and M • − σ 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.

Some Predictions of Scaling Relations: The Case of the Black Hole in M87

“…This correlation was tested with many different samples and fitting methods (Feoli and Mele, 2007;Feoli and Mancini, 2009;Mancini and Feoli, 2012;Benedetto et al 2013;Iannella and Feoli, 2020) and a theoretical background was proposed in Feoli (2014b). Furthermore, in Iannella et al (2021), the predictive power of the relation has been analysed and the statistical elaboration done previously has been enhanced. Finally, this relation proved to be very competitive with its very low intrinsic scatter (Saglia et al 2016).…”