A unifying representation of some case-deletion influence measures in univariate and multivariate linear regression

Abstract

In recent years numerous measures have been proposed for assessing the influence of observations on least-squares regression results. Various influence measures were introduced based on different motivational arguments and each was designed to measure the influence of observations on different aspects of various regression results. The chief motivation and result of this paper is the determination of the main mathematical components common to many multiple-case regression diagnostics, and to make use of these components in formulating an algebraic representation which unifies the existing measures. The representation, which we refer to as the J I -class, is based on a few common interpretable components. These components are functions of the elements of two orthogonal projector matrices; the projector matrix for the column space of the explanatory variables, and the projector matrix for the residuals. The proposed representation enables one to perceive such existing measures not only in their original diversified formulations, but also in a manner that allows associations or similarities to be recognized among them in a clear, concise fashion. The class possesses several properties that can be used to study the relationships among the existing influence measures, especially those with seemingly different motivations and initial characterizations. The J I -class, which we use for a unifying representation of some existing influence measures, can also be used to generate new influence measures.