Authors: Zhu, Li-Ping¹; Zhu, Li-Xing²

Source: Journal of the Royal Statistical Society: Series B (Statistical Methodology), Volume 71, Number 2, April 2009 , pp. 525-548(24)

Publisher: Wiley-Blackwell

Abstract:

Summary. 

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^1/2/ log (n)} and o(n^1/3) respectively where n is the sample size. When there are no other constraints on the covariance of predictors, the rates n^1/2 and n^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.

Keywords: Central subspace; Collinearity; Distribution function; Inverse regression; Least squares estimation; Partial least squares

Document Type: Research article

DOI: http://dx.doi.org/10.1111/j.1467-9868.2008.00697.x

Affiliations: 1: East China Normal University, Shanghai, People's Republic of China 2: East China Normal University, Shanghai, and Hong Kong Baptist University, People's Republic of China

Publication date: 2009-04-01

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