On distribution-weighted partial least squares with diverging number of highly correlated predictors

Li-Ping Zhu and Li-Xing Zhu

Journal Of The Royal Statistical Society Series B, 2009, vol. 71, issue 2, pages 525-548

Abstract: Because highly correlated data arise from many scientific fields, we investigate parameter estimation in a semiparametric regression model with diverging number of predictors that are highly correlated. For this, we first develop a distribution-weighted least squares estimator that can recover directions in the central subspace, then use the distribution-weighted least squares estimator as a seed vector and project it onto a Krylov space by partial least squares to avoid computing the inverse of the covariance of predictors. Thus, distrbution-weighted partial least squares can handle the cases with high dimensional and highly correlated predictors. Furthermore, we also suggest an iterative algorithm for obtaining a better initial value before implementing partial least squares. For theoretical investigation, we obtain strong consistency and asymptotic normality when the dimension "p" of predictors is of convergence rate "O"{"n"-super-1/2/ log ("n")} and "o"("n"-super-1/3) respectively where "n" is the sample size. When there are no other constraints on the covariance of predictors, the rates "n"-super-1/2 and "n"-super-1/3 are optimal. We also propose a Bayesian information criterion type of criterion to estimate the dimension of the Krylov space in the partial least squares procedure. Illustrative examples with a real data set and comprehensive simulations demonstrate that the method is robust to non-ellipticity and works well even in 'small "n"-large "p"' problems. Copyright (c) 2009 Royal Statistical Society.

Date: 2009

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