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April 2012 Perturbation and scaled Cook’s distance
Hongtu Zhu, Joseph G. Ibrahim, Hyunsoon Cho
Ann. Statist. 40(2): 785-811 (April 2012). DOI: 10.1214/12-AOS978


Cook’s distance [Technometrics 19 (1977) 15–18] is one of the most important diagnostic tools for detecting influential individual or subsets of observations in linear regression for cross-sectional data. However, for many complex data structures (e.g., longitudinal data), no rigorous approach has been developed to address a fundamental issue: deleting subsets with different numbers of observations introduces different degrees of perturbation to the current model fitted to the data, and the magnitude of Cook’s distance is associated with the degree of the perturbation. The aim of this paper is to address this issue in general parametric models with complex data structures. We propose a new quantity for measuring the degree of the perturbation introduced by deleting a subset. We use stochastic ordering to quantify the stochastic relationship between the degree of the perturbation and the magnitude of Cook’s distance. We develop several scaled Cook’s distances to resolve the comparison of Cook’s distance for different subset deletions. Theoretical and numerical examples are examined to highlight the broad spectrum of applications of these scaled Cook’s distances in a formal influence analysis.


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Hongtu Zhu. Joseph G. Ibrahim. Hyunsoon Cho. "Perturbation and scaled Cook’s distance." Ann. Statist. 40 (2) 785 - 811, April 2012.


Published: April 2012
First available in Project Euclid: 17 May 2012

zbMATH: 1273.62180
MathSciNet: MR2933666
Digital Object Identifier: 10.1214/12-AOS978

Primary: 62J20

Keywords: conditionally scaled Cook’s distance , Cook’s distance , perturbation , relative influential , scaled Cook’s distance , size issue

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 2 • April 2012
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