The Derivative of Influence Function, Location Breakdown Point, Group Leverage and Regression Residuals' Plots

Abstract

In several linear regression data sets, $Y (\in R)$ on ${\bf X} (\in R^p),$ visual comparisons of $L_1$ and $L_2$-residuals' plots indicate bad leverage cases. The phenomenon is confirmed theoretically by introducing Location Breakdown Point (LBP) of a functional $T$: any point where the derivative of $T$'s Influence Function either takes values at infinities or does not exist. Guidelines for the plots' visual comparisons as diagnostic are provided. The new tools used include E-matrix and suggest influence diagnostic RINFIN which measures the distance in the derivatives of $L_2$-residuals} at $({\bf x},y)$ from model $F$ and from gross-error model $F_{\epsilon, {\bf x},y}.$ The larger RINFIN$({\bf x},y)$ is, the larger $({\bf x},y)$'s influence in $L_2$-regression residual is. RINFIN allows measuring group influence of $k$ ${\bf x}$-neighboring data cases in a size $n$ sample using their average, $(\bar {\bf x}_k,\bar y_k),$ as one case with weight $\epsilon=k/n.$ For high dimensional, simulated data, the misclassification proportion of bad leverage cases in data's RINFIN-ordering decreases to zero as $p$ increases, thus reconfirming the blessing of high dimensionality in the detection of remote clusters. The visual diagnostic and RINFIN are successful in applications and complement each other.