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- Title
On distribution-weighted partial least squares with diverging number of highly correlated predictors.
- Authors
Li-Ping Zhu; Li-Xing Zhu
- 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{ n1/2/ log ( n)} and o( n1/3) respectively where n is the sample size. When there are no other constraints on the covariance of predictors, the rates n1/2 and n1/3 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.
- Subjects
REGRESSION analysis; LEAST squares; ALGORITHMS; STOCHASTIC convergence; MATHEMATICAL statistics; ALGEBRA
- Publication
Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2009, Vol 71, Issue 2, p525
- ISSN
1369-7412
- Publication type
Article
- DOI
10.1111/j.1467-9868.2008.00697.x