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Biometrika Advance Access originally published online on August 5, 2007
Biometrika 2007 94(3):615-625; doi:10.1093/biomet/asm043
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Copyright © 2007 Biometrika Trust

Articles

Partial inverse regression

Lexin Li

Department of Statistics, North Carolina State University, Raleigh, North Carolina 27695, U.S.A.

R. Dennis Cook

School of Statistics, University of Minnesota, St Paul, Minnesota 55108, U.S.A.

Chih-Ling Tsai

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.
  Abstract

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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