Perturbation selection and influence measures in local influence analysis

TL;DR

This paper develops a differential-geometrical framework for local influence analysis, addressing perturbation selection and influence measures for objective functions at any point, with applications to regression models.

Contribution

It introduces a perturbation manifold and associated metrics to improve perturbation selection and influence measurement in local influence analysis.

Findings

Metric tensor guides perturbation choice

New influence measures applicable at any point

Effective identification of influential observations in models

Abstract

Cook's [J. Roy. Statist. Soc. Ser. B 48 (1986) 133--169] local influence approach based on normal curvature is an important diagnostic tool for assessing local influence of minor perturbations to a statistical model. However, no rigorous approach has been developed to address two fundamental issues: the selection of an appropriate perturbation and the development of influence measures for objective functions at a point with a nonzero first derivative. The aim of this paper is to develop a differential--geometrical framework of a perturbation model (called the perturbation manifold) and utilize associated metric tensor and affine curvatures to resolve these issues. We will show that the metric tensor of the perturbation manifold provides important information about selecting an appropriate perturbation of a model. Moreover, we will introduce new influence measures that are applicable to objective functions at any point. Examples including linear regression models and linear mixed models are examined to demonstrate the effectiveness of using new influence measures for the identification of influential observations.