Department of Mathematics
On distribution-weighted partial least squares with diverging number of highly correlated predictors
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. © 2009 Royal Statistical Society.
Central subspace, Collinearity, Distribution function, Inverse regression, Least squares estimation, Partial least squares
Source Publication Title
Journal of the Royal Statistical Society: Series B
Royal Statistical Society
Link to Publisher's Edition
Zhu, Li-Ping, and Li-Xing Zhu. "On distribution-weighted partial least squares with diverging number of highly correlated predictors." Journal of the Royal Statistical Society: Series B 71.2 (2009): 525-548.