Biometrika Advance Access originally published online on August 5, 2007
Biometrika 2007 94(3):585-601; doi:10.1093/biomet/asm055
Copyright © 2007 Biometrika Trust
Articles |
Implications of influence function analysis for sliced inverse regression and sliced average variance estimation
Department of Mathematics and Statistics, La Trobe University, Melbourne 3086, Australia
luke.prendergast{at}latrobe.edu.au
Received for publication 1 May 2006. Revision received 1 February 2007.
Sliced inverse regression, sliced inverse regression II and sliced average variance estimation are three related dimension-reduction methods that require relatively mild model assumptions. As an approximation for the relative influence of single observations from large samples, the influence function is used to compare the sensitivity of the three methods to particular observational types. The analysis carried out here helps to explain why there is a lack of agreement concerning the preferability of these dimension-reduction procedures in general. An efficient sample version of the influence function is also developed and evaluated.
Key Words: Bénasséni's coefficient • Dimension reduction • Influence function • Robustness • Sliced average variance estimation • Sliced inverse regression
References
-
Bénasséni J. Sensitivity coefficients for the subspaces spanned by principal components. Commun. Statist. A (1990) 19:2021–34.
Chen C.-H., Li K.-C. Can SIR be as popular as multiple linear regression? Statist. Sinica (1998) 8:289–316.
Cook R. D. Detection of influential observation in linear regression. Technometrics (1977) 19:15–8.[CrossRef][Web of Science]
Cook R. D. Regression Graphics: Ideas for Studying Regressions Through Graphics (1998) New York: Wiley.
Cook R. D., Weisberg S. Discussion of a paper by K.-C. Li. J. Am. Statist. Assoc. (1991) 86:328–32.[CrossRef][Web of Science]
Critchley F. Influence in principal components analysis. Biometrika (1985) 72:627–36.
Fearn T. A misuse of ridge regression in the calibration of near infrared reflectance instruments. Appl. Statist. (1983) 32:73–9.[CrossRef]
Gather U., Hilker T., Becker C. A Robustified version of sliced inverse regression. In: Statistics in Genetics and in the Environmental Sciences—Fernholz L. T., Morgenthaler S., Stahel W., eds. (2001) Birkhäuser: Basel. 147–57.
Gather U., Hilker T., Becker C. A note on outlier sensitivity of sliced inverse regression. Statistics (2002) 13:271–81.
Hall P., Li K.-C. On almost linearity of low dimensional projections from high dimensional data. Ann. Statist. (1993) 21:867–89.
Hampel F. The influence curve and its role in robust estimation. J. Am. Statist. Assoc. (1974) 69:383–93.[CrossRef][Web of Science]
Härdle W., Tsybakov A. B. Discussion of a paper by K.-C. Li. J. Am. Statist. Assoc. (1991) 86:333–5.
Harrison D., Rubinfeld D. L. Hedonic housing prices and the demand for clean air. J. Environ. Econ. Manag. (1978) 5:81–102.[CrossRef]
Horn R. A., Johnson C. R. Matrix Analysis (1985) Cambridge: Cambridge University Press.
Kent J. T. Discussion of a paper by K.-C. Li. J. Am. Statist. Assoc. (1991) 86:336–7.[CrossRef][Web of Science]
Li K.-C. Sliced inverse regression for dimension reduction (with Discussion). J. Am. Statist. Assoc. (1991a) 86:316–42.[CrossRef][Web of Science]
Li K.-C. Rejoinder to the discussion of Li (1991a). J. Am. Statist. Assoc. (1991b) 86:337–42.[CrossRef][Web of Science]
Prendergast L. A. Influence functions for sliced inverse regression. Scand. J. Statist. (2005) 32:385–404.[CrossRef]
Prendergast L. A. Detecting influential observations on the SIR e.d.r. space. Aust. New Zeal. J. Statist. (2006) 48:285–304.[CrossRef]
Welsh A. H. Discussion of a paper by R. D. Cook and X. Yin. Aust. New Zeal. J. Statist. (2001) 43:177–9.
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