Wu Yichao, Li Lexin
Department of Statistics, North Carolina State University, Raleigh, NC 27695, USA.
Stat Sin. 2011;2011(21):707-730. doi: 10.5705/ss.2011.031a.
We investigate asymptotic properties of a family of sufficient dimension reduction estimators when the number of predictors p diverges to infinity with the sample size. We adopt a general formulation of dimension reduction estimation through least squares regression of a set of transformations of the response. This formulation allows us to establish the consistency of reduction projection estimation. We then introduce the SCAD max penalty, along with a difference convex optimization algorithm, to achieve variable selection. We show that the penalized estimator selects all truly relevant predictors and excludes all irrelevant ones with probability approaching one, meanwhile it maintains consistent reduction basis estimation for relevant predictors. Our work differs from most model-based selection methods in that it does not require a traditional model, and it extends existing sufficient dimension reduction and model-free variable selection approaches from the fixed p scenario to a diverging p.
我们研究了一类充分降维估计量的渐近性质，此时预测变量的数量(p)随着样本量趋于无穷大。我们通过对响应变量的一组变换进行最小二乘回归，采用了一种通用的降维估计公式。这种公式使我们能够建立降维投影估计的一致性。然后，我们引入SCAD最大惩罚项以及一种差分凸优化算法来实现变量选择。我们表明，惩罚估计量以概率趋近于1选择所有真正相关的预测变量并排除所有不相关的预测变量，同时它对相关预测变量保持一致的降维基估计。我们的工作与大多数基于模型的选择方法不同，因为它不需要传统模型，并且将现有的充分降维及无模型变量选择方法从固定(p)的情形扩展到了(p)发散的情形。
