On distribution-weighted partial least squares with diverging number of highly correlated predictors |
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Authors: | Li-Ping Zhu Li-Xing Zhu |
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Affiliation: | East China Normal University, Shanghai, People's Republic of China; East China Normal University, Shanghai, and Hong Kong Baptist University, People's Republic of China |
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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. |
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Keywords: | Central subspace Collinearity Distribution function Inverse regression Least squares estimation Partial least squares |
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