基于Bootstrap的高维非线性模型渐近精化方法
Bootstrap based asymptotic refinements for high-dimensional nonlinear models
AI总结：
本文研究高维稀疏非线性模型的SCAD惩罚估计，提出基于阈值化SCAD估计的bootstrap方法，证明其能为对称检验提供O(n^-2)、为单侧或等尾检验提供O(n^-1)的渐近精化，并通过蒙特卡洛实验验证其显著降低拒绝与覆盖概率误差。
摘要：
我们考虑高维、可能非线性的模型的惩罚极值估计，该模型具有稀疏性，即其大部分参数为零，但部分参数不为零。我们使用SCAD惩罚函数，在适当条件下该函数可提供模型选择一致且具有oracle效率的估计。然而，基于oracle模型的渐近近似在许多应用中所遇到的样本量下可能不够准确。本文给出了相应条件，在这些条件下，基于通过带阈值化的SCAD惩罚所获得估计的bootstrap方法，能够为对称假设检验（置信区间）的拒绝（覆盖）概率误差提供阶为 \\(O \left( n^{- 2} \right)\\) 的渐近精化，并为单侧或等尾检验（置信区间）的拒绝（覆盖）概率误差提供阶为 \\(O \left( n^{- 1} \right)\\) 的渐近精化。蒙特卡洛实验结果表明，bootstrap能够大幅降低拒绝概率和覆盖概率的误差。即使某些参数接近但不等于零，bootstrap也是一致的，尽管它不一定提供渐近精化。随机系数logit和probit模型以及非线性矩模型是该程序适用模型的例子。
英文摘要：
We consider penalized extremum estimation of a high-dimensional, possibly nonlinear model that is sparse in the sense that most of its parameters are zero but some are not. We use the SCAD penalty function, which provides model selection consistent and oracle efficient estimates under suitable conditions. However, asymptotic approximations based on the oracle model can be inaccurate with the sample sizes found in many applications. This paper gives conditions under which the bootstrap, based on estimates obtained through SCAD penalization with thresholding, provides asymptotic refinements of size \(O \left( n^{- 2} \right)\) for the error in the rejection (coverage) probability of a symmetric hypothesis test (confidence interval) and \(O \left( n^{- 1} \right)\) for the error in the rejection (coverage) probability of a one-sided or equal tailed test (confidence interval). The results of Monte Carlo experiments show that the bootstrap can provide large reductions in errors in rejection and coverage probabilities. The bootstrap is consistent, though it does not necessarily provide asymptotic refinements, even if some parameters are close but not equal to zero. Random-coefficients logit and probit models and nonlinear moment models are examples of models to which the procedure applies.
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