广义部分线性单指标模型的最优模型平均方法  

Optimal model averaging for generalized partially linear single-index models

作　　者：张建强 陈泽 廖军 许王莉 Jianqiang Zhang;Ze Chen;Jun Liao;Wangli Xu

机构地区：[1]中国人民大学应用统计科学研究中心,北京100872 [2]中国人民大学统计学院,北京100872 [3]山东大学中泰证券金融研究院,济南250100

出　　处：《中国科学：数学》2023年第11期1461-1486,共26页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11971478,12001534和11971323);北京市自然科学基金重点研究专项(批准号:Z200001);中国人民大学公共健康与疾病预防控制交叉学科重大创新平台(批准号:2021PDPC)资助项目。

摘　　要：本文研究广义部分线性单指标模型(generalized partially linear single-index models,GPLSIMs)的模型平均问题.在实际应用中,GPLSIMs由于其灵活性和易解释性受到广泛关注.然而,GPLSIMs在应用中存在两类不确定性:变量的不确定性和单指标连接函数光滑度的不确定性.为了解决该不确定性问题,本文提出一种GPLSIMs的最优模型平均方法,该方法通过最大交叉验证准则得到数据驱动的权重.在模型误设定假设和发散模型空间的框架下,本文证明在最小化Kullback-Leibler(KL)损失准则下,所提出的模型平均估计渐近最优.同时,当候选模型集中存在伪真模型时,本文证明基于交叉验证准则得到的权重渐近地集中在伪真模型上.此外,基于提出的模型平均方法,本文为GPLSIMs构建了一种变量重要性度量,并证明该度量可以渐近识别所有真实模型中的变量.模拟研究和两个实际数据分析均展示了本文提出的方法相对于几种现有方法的优势.This paper concerns model averaging for generalized partially linear single-index models(GPLSIMs).GPLSIMs have attracted increasing attention due to their great flexibility and easy interpretability.However,it is tough to determine the predictors and the degree of smoothing in the single-index link function in practice.To deal with this model selection uncertainty problem,we propose an optimal model averaging approach for GPLSIMs,in which we select the data-driven weights by maximizing a cross-validation criterion.Under model misspecification,we prove that the resulting model averaging estimator is asymptotically optimal in the sense of achieving the lowest possible Kullback-Leibler loss in the framework of diverging model spaces.On the other hand,when pseudo-correct models are available in the candidate model set,we demonstrate that the cross-validation scheme asymptotically assigns all weights to the pseudo-correct models.Moreover,based on our model averaging approach,we develop a reliable variable importance measure for GPLSIMs,which can be shown to asymptotically identify all the variables in the true model.Simulation experiments and two empirical applications illustrate the superiority of the proposed method over several existing methods.

关 键 词：广义部分线性单指标模型 交叉验证 渐近最优性 权重一致性 变量重要性 

分 类 号：TP3[自动化与计算机技术—计算机科学与技术]