Biometrika Advance Access originally published online on August 5, 2007
Biometrika 2007 94(3):615-625; doi:10.1093/biomet/asm043
Copyright © 2007 Biometrika Trust
Articles |
Partial inverse regression
Department of Statistics, North Carolina State University, Raleigh, North Carolina 27695, U.S.A.
School of Statistics, University of Minnesota, St Paul, Minnesota 55108, U.S.A.
Graduate School of Management, University of California, Davis, California 95616, U.S.A.
li{at}stat.ncsu.edu
dennis{at}stat.umn.edu
cltsai{at}ucdavis.edu
Received for publication 1 March 2005. Revision received 1 December 2006.
In regression with a vector of quantitative predictors, sufficient dimension reduction methods can effectively reduce the predictor dimension, while preserving full regression information and assuming no parametric model. However, all current reduction methods require the sample size n to be greater than the number of predictors p. It is well known that partial least squares can deal with problems with n < p. We first establish a link between partial least squares and sufficient dimension reduction. Motivated by this link, we then propose a new dimension reduction method, entitled partial inverse regression. We show that its sample estimator is consistent, and that its performance is similar to or superior to partial least squares when n < p, especially when the regression model is nonlinear or heteroscedastic. An example involving the spectroscopy analysis of biscuit dough is also given.
Key Words: Partial least squares • Single-index model • Sliced inverse regression
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