SEMIPARAMETRIC ESTIMATION AND VARIABLE SELECTION FOR SPARSE SINGLE INDEX MODELS IN INCREASING DIMENSION
研究了高维单指标模型中半参数筛估计，利用Hermite多项式逼近未知连接函数，通过惩罚加权线性回归估计指标参数，并提出了前向回归筛选方法用于超高维变量选择。
This paper considers semiparametric sieve estimation in high-dimensional single index models. The use of Hermite polynomials in approximating the unknown link function provides a convenient framework to conduct both estimation and variable selection. The estimation of the index parameter is formulated from solutions obtained by the routine penalized weighted linear regression procedure, where the weights are used in order to tackle the unbounded support of the regressors. The resulting index parameter estimator is shown to be consistent and sparse, and the asymptotic normality for the estimators of both the index parameter and the link function is established. To perform variable selection in the ultra-high dimension case, we further suggest a forward regression screening method, which is shown to enjoy the sure independence screening property. This screening procedure can be used before the penalized variable selection to reduce the burden of dimensionality. Numerical results show that both the variable selection procedures and the associated estimators perform well in finite samples.