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Influence analysis based on the case sensitivity function

Author

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  • Frank Critchley
  • Richard A. Atkinson
  • Guobing Lu
  • Elenice Biazi

Abstract

The case sensitivity function approach to influence analysis is introduced as a natural smooth extension of influence curve methodology in which both the insights of geometry and the power of (convex) analysis are available. In it, perturbation is defined as movement between probability vectors defining weighted empirical distributions. A Euclidean geometry is proposed giving such perturbations both size and direction. The notion of the salience of a perturbation is emphasized. This approach has several benefits. A general probability case weight analysis results. Answers to a number of outstanding questions follow directly. Rescaled versions of the three usual finite sample influence curve measures—seen now to be required for comparability across different‐sized subsets of cases—are readily available. These new diagnostics directly measure the salience of the (infinitesimal) perturbations involved. Their essential unity, both within and between subsets, is evident geometrically. Finally it is shown how a relaxation strategy, in which a high dimensional (O(nCm)) discrete problem is replaced by a low dimensional (O(n)) continuous problem, can combine with (convex) optimization results to deliver better performance in challenging multiple‐case influence problems. Further developments are briefly indicated.

Suggested Citation

  • Frank Critchley & Richard A. Atkinson & Guobing Lu & Elenice Biazi, 2001. "Influence analysis based on the case sensitivity function," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 307-323.
  • Handle: RePEc:bla:jorssb:v:63:y:2001:i:2:p:307-323
    DOI: 10.1111/1467-9868.00287
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    Cited by:

    1. Shi, Lei & Huang, Mei, 2011. "Stepwise local influence analysis," Computational Statistics & Data Analysis, Elsevier, vol. 55(2), pages 973-982, February.
    2. Jun Lu & Wen Gan & Lei Shi, 2022. "Local influence analysis for GMM estimation," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 106(1), pages 1-23, March.
    3. Vasconcellos, Klaus L.P. & Zea Fernandez, L.M., 2009. "Influence analysis with homogeneous linear restrictions," Computational Statistics & Data Analysis, Elsevier, vol. 53(11), pages 3787-3794, September.
    4. Lee, Sik-Yum & Xu, Liang, 2004. "Influence analyses of nonlinear mixed-effects models," Computational Statistics & Data Analysis, Elsevier, vol. 45(2), pages 321-341, March.
    5. Marc G. Genton & Peter Hall, 2016. "A tilting approach to ranking influence," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(1), pages 77-97, January.
    6. Xibin Zhang & Maxwell L. King, 2002. "Influence Diagnostics in GARCH Processes," Monash Econometrics and Business Statistics Working Papers 19/02, Monash University, Department of Econometrics and Business Statistics.
    7. Peter Hall & D. M. Titterington & Jing‐Hao Xue, 2009. "Tilting methods for assessing the influence of components in a classifier," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(4), pages 783-803, September.

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