近似稀疏高维非参与半参模型的Dantzig Selector的研究

Sparse high-dimensional models have been widely studied and applied, there have been many useful variable selection methods which turn the high-dimensional models into low applicable models, but there are little literatures about the nearly spare high-dimensional models. Due to the limitations of sparsity conditions, our project mainly study some high-dimensional nonparametric and semi-parametric models under nearly sparse conditions. The main contents are as follows: firstly, to construct the variable selection theory and method for nearly sparse high-dimensional nonparametric and semi-parametric models, that is, we first transform the nonparametric and semi-parametric models into the linear structure of paprameters, then we consider the model selection consistency and parameters estimation consistency of dantzig selector under nearly sparsity assumption; secondly, sub-models after variable selection must be biased because the model is not sparse but nearly sparse, so it is necessary to justify the biased model by some artificial constitute to eliminate or at least reduce the bias; thirdly, for the justified sub-models, we construct the parameters and nonparameters consistent estimations, and the justification of the sub-models is very helpful for improvement of the model prediction accuracy. Since the nearly sparsity assumption is very reasonable in applications, it is very expectable that our study is innovative in theory and valuable in application.

稀疏条件下的高维统计模型已有广泛深入的研究和应用，但稀疏性在应用中具有局限性，很多情况下大多数变量的系数很小并非严格为零，我们称之为近似稀疏。文献中对近似稀疏条件下高维模型的研究非常少， 故本项目致力于近似稀疏条件下的高维非参与半参模型的研究。主要研究内容为：一是建立近似稀疏条件下高维非参与半参数模型的变量选择的理论和方法，即对我们所要研究的非参与半参数模型进行适当的变形，变为参数的线性结构后，研究其在近似稀疏条件下dantzig selector变量选择方法的模型选择相合性及参数估计的相合性；二是由于变量选择后的子模型在近似稀疏的假设下必有偏，故选择适当的备选结构，对子模型进行纠偏调整，使之具有无偏性；三是对于纠偏调整后的子模型构造参数和非参的相合估计，从而利用调整子模型进行预测，提高预测精度。由于近似稀疏性假设在实际应用中的存在合理性和普遍性，从而可以期待,本研究理论上有所创新。

本项目主要研究Dantzig selector的高维变量选择的大样本性质， 以及进一步在近似稀疏条件下,高维模型变量选择的渐进性质，以及模型选择后有偏子模型的纠偏问题。经过项目组三年的研究工作，我们在建立Dantzig selector以及adaptive Dantzig selector方法在高维线性模型的模型选择相合性, 参数估计的相合性；对高维条件下单指标模型，变换模型等的dantzig selector 变量选择的大样本性质，以及在近似稀疏条件下变量选择的相合性和子模型的纠偏问题等方面，得到了一系列的研究结果。项目基本按照计划书开展各项工作，达到了预期的目标。