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.

翻译：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光滑样条估计量的Bahadur表示，并利用该结果证明了在给定点处指标链接函数的估计量与参数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：尽管单指标模型中链接函数是在指标系数的线性组合处求值的，但链接函数的估计量与指标系数的估计量是渐近独立的。我们的方法利用了基于再生核希尔伯特空间和经验过程理论的精细分析。我们证明了光滑样条估计量在$L^2$风险下达到了极小极大最优速率，并考虑了多个需要对所有模型参数进行联合推断的统计应用。特别地，我们为链接函数构建了同步置信带，并提出了推断工具以研究（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。我们还构建了涉及非参数和参数分量的模型参数联合假设检验，并提出了新颖的乘子自助法程序以避免估计未知的渐近量。