Partial Dynamic Dimension Reduction for Conditional Mean in Regression
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Partial Dynamic Dimension Reduction for Conditional Mean in Regression∗ GAN Shengjin · YU Zhou
DOI: 10.1007/s11424-020-8329-3 Received: 23 November 2018 / Revised: 22 November 2019 c The Editorial Office of JSSC & Springer-Verlag GmbH Germany 2020 Abstract In many regression analysis, the authors are interested in regression mean of response variate given predictors, not its the conditional distribution. This paper is concerned with dimension reduction of predictors in sense of mean function of response conditioning on predictors. The authors introduce the notion of partial dynamic central mean dimension reduction subspace, different from central mean dimension reduction subspace, it has varying subspace in the domain of predictors, and its structural dimensionality may not be the same point by point. The authors study the property of partial dynamic central mean dimension reduction subspace, and develop estimated methods called dynamic ordinary least squares and dynamic principal Hessian directions, which are extension of ordinary least squares and principal Hessian directions based on central mean dimension reduction subspace. The kernel estimate methods for dynamic ordinary least squares and dynamic Principal Hessian Directions are employed, and large sample properties of estimators are given under the regular conditions. Simulations and real data analysis demonstrate that they are effective. Keywords Dynamic ordinary least square estimate, dynamic principal Hessian directions, kernel estimate, partial dimension reduction.
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Introduction
The goal of regression analysis is to find how response varying with predictors, including studying for distribution of response conditioning on predictors and regression mean function of response on predictors. However, when the dimensionality of predictors is high, especially in big data times, it may occur curse of dimension in data analysis, one that tackles the GAN Shengjin School of Electronical and Information Engineering, Fuqing Branch of Fujian Normal University, Fuqing 350300, China. Email: [email protected]. YU Zhou (Corresponding author) School of Statistics, East China Normal University, Shanghai 200062, China. Email: [email protected]. ∗ This research was supported by the Natural Science Foundation of Fujian Province of China under Grant No. 2018J01662, and High-Level Cultivation Project of Fuqing Branch of Fujian Normal University under Grant No. KY2018S02. This paper was recommended for publication by Editor SUN Liuquan.
GAN SHENGJIN · YU ZHOU
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problem is dimension reduction. Sufficient dimension reduction sakes few linear combinations of predictors to character the distribution of response conditioning on predictors without loss of information of predictors about response. Specifically, let response Y be a variate and predictors X = (x1 , x2 , · · · , xp )T with dimensionality p, if there exists a matrix η ∈ Rp×d (d ≤ p) satisfying the follow formula |=
X|η T X,
(1)
|=
Y
stands for statistical independency. Formula (1) means that the distributi
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