超高维部分线性单指标模型中的估计与推断
Estimation and Inference in Ultrahigh Dimensional Partially Linear Single-Index Models
AI总结：
针对超高维部分线性单指标模型，提出轮廓部分惩罚最小二乘估计及参数F型检验与非参数函数设定检验，并通过模拟和实际数据验证其性能。
摘要：
本文关注超高维部分线性单指标模型的估计与推断。高维冗余参数与冗余未知函数的存在，使估计和推断问题极具挑战性。本文首先提出一种轮廓部分惩罚最小二乘估计量，并在超高维设定下建立该估计量的稀疏性、一致性和渐近表示。随后，针对主要感兴趣的参数提出一个F型检验统计量，证明其在原假设下的极限分布为χ²分布，并且该检验统计量能够检测以root-n速率收敛于原假设的局部备择假设。本文进一步针对非参数函数的设定检验问题提出一种新检验，并证明该检验统计量渐近服从正态分布。通过模拟研究考察所提估计量和检验的有限样本表现，并用一个实际数据示例说明所提方法流程。
英文摘要：
This paper is concerned with estimation and inference for ultrahigh dimensional partially linear single-index models. The presence of high dimensional nuisance parameter and nuisance unknown function makes the estimation and inference problem very challenging. In this paper, we first propose a profile partial penalized least squares estimator and establish the sparsity, consistency and asymptotic representation of the proposed estimator in ultrahigh dimensional setting. We then propose an $F$-type test statistic for parameters of primary interest and show that the limiting null distribution of the test statistic is $χ^2$ distribution, and the test statistic can detect local alternatives, which converge to the null hypothesis at the root-$n$ rate. We further propose a new test for the specification testing problem of the nonparametric function. The test statistic is shown to be asymptotically normal. Simulation studies are conducted to examine the finite sample performance of the proposed estimators and tests. A real data example is used to illustrate the proposed procedures.
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