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

TL;DR

This paper compares two black hole-host galaxy scaling relations, emphasizing predictive accuracy and residual analysis to better understand their statistical properties and physical implications.

Contribution

It introduces a focus on predictive accuracy via cross-validation and residual analysis to improve interpretation of the $M_{\bullet}-M_{G}\sigma^2$ relation.

Findings

Predictive accuracy varies across samples.

Residual analysis reveals variable influences.

Comparison clarifies relation robustness.

Abstract

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.