Open Access
December 2007 Perturbation selection and influence measures in local influence analysis
Hongtu Zhu, Joseph G. Ibrahim, Sikyum Lee, Heping Zhang
Ann. Statist. 35(6): 2565-2588 (December 2007). DOI: 10.1214/009053607000000343


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.


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Hongtu Zhu. Joseph G. Ibrahim. Sikyum Lee. Heping Zhang. "Perturbation selection and influence measures in local influence analysis." Ann. Statist. 35 (6) 2565 - 2588, December 2007.


Published: December 2007
First available in Project Euclid: 22 January 2008

zbMATH: 1129.62068
MathSciNet: MR2382658
Digital Object Identifier: 10.1214/009053607000000343

Primary: 62J20
Secondary: 62-07

Keywords: Appropriate perturbation , linear mixed models , Linear regression , local influence , perturbation manifold

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 6 • December 2007
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