Title: Simultaneous semiparametric inference for single-index models Show Chinese title

Title: 同时半参数单指标模型推断

Abstract: In the common partially linear single-index model we establish a Bahadur representation for a smoothing spline estimator of all model parameters and use this result to prove the joint weak convergence of the estimator of the index link function at a given point, together with the estimators of the parametric regression coefficients. We obtain the surprising result that, despite of the nature of single-index models where the link function is evaluated at a linear combination of the index-coefficients, the estimator of the link function and the estimator of the index-coefficients are asymptotically independent. Our approach leverages a delicate analysis based on reproducing kernel Hilbert space and empirical process theory. We show that the smoothing spline estimator achieves the minimax optimal rate with respect to the $L^2$-risk and consider several statistical applications where joint inference on all model parameters is of interest. In particular, we develop a simultaneous confidence band for the link function and propose inference tools to investigate if the maximum absolute deviation between the (unknown) link function and a given function exceeds a given threshold. We also construct tests for joint hypotheses regarding model parameters which involve both the nonparametric and parametric components and propose novel multiplier bootstrap procedures to avoid the estimation of unknown asymptotic quantities. Show Chinese abstract

Abstract: 在常见的部分线性单指标模型中，我们建立了所有模型参数的平滑样条估计量的Bahadur表示，并利用该结果证明了在给定点处指数链接函数估计量与回归系数估计量的联合弱收敛性。 我们得到了一个令人惊讶的结果：尽管单指标模型的链接函数是在指数系数的线性组合上进行评估的，但链接函数的估计量和指数系数的估计量在渐近意义上是独立的。 我们的方法基于再生核Hilbert空间和经验过程理论的精细分析。 我们表明，平滑样条估计量在$L^2$-风险下达到了最小极大最优速率，并考虑了几种统计应用，其中对所有模型参数的联合推断是有意义的。 特别是，我们开发了链接函数的同时置信带，并提出了推断工具以调查未知链接函数与给定函数之间的最大绝对偏差是否超过给定阈值。 我们还构建了关于涉及非参数和参数成分的模型参数联合假设检验，并提出了新颖的倍乘自助法程序以避免估计未知的渐近量。