Under the reproducing kernel Hilbert spaces (RKHS), we consider the penalized least-squares of the partially functional linear models (PFLM), whose predictor contains both functional and traditional multivariate parts, and the multivariate part allows a divergent number of parameters. From the non-asymptotic point of view, we focus on the rate-optimal upper and lower bounds of the prediction error. An exact upper bound for the excess prediction risk is shown in a non-asymptotic form under a more general assumption known as the effective dimension to the model, by which we also show the prediction consistency when the number of multivariate covariates $p$ slightly increases with the sample size $n$. Our new finding implies a trade-off between the number of non-functional predictors and the effective dimension of the kernel principal components to ensure prediction consistency in the increasing-dimensional setting. The analysis in our proof hinges on the spectral condition of the sandwich operator of the covariance operator and the reproducing kernel, and on sub-Gaussian and Berstein concentration inequalities for the random elements in Hilbert space. Finally, we derive the non-asymptotic minimax lower bound under the regularity assumption of the Kullback-Leibler divergence of the models.
Joint asymptotics for semi-nonparametric regression models with partially linear structure
我们考虑一个联合渐近框架研究半非参数回归模型，其中（有限维）欧几里德参数和（无限维）功能参数都感兴趣。考虑的模型类共享一个部分线性结构，并在两个一般情况下估计：（i）准似然和（ii）真似然。我们首先证明了欧氏估计和（逐点）功能估计，这是重新缩放在不同的速率，共同收敛到一个零均值高斯向量。这个弱收敛结果揭示了一个令人惊讶的联合渐近现象：这两个估计是渐近独立的。本文的一个主要目标是获得第一手的洞察上述现象。此外，一个似然比测试提出了一组联合本地的假设，其中一个新版本的威尔克斯现象[安。9（1938）60-62; Ann. Statistist. 1（2001）153-193]已公布。一种新的技术工具，称为联合Bahadur表示，开发研究这些联合渐近的结果。
Parametric estimation for partially hidden diffusion processes sampled at discrete times
对于一个一维扩散过程$X=\{X(t) ; 0\leq t \leq T \}$，我们假设如果$X(t)$低于某个固定且已知的阈值$τ$，则它是隐藏的，否则它是可见的。这意味着一个部分隐藏的扩散过程。本文讨论的问题是在离散抽样方案得到的部分隐扩散过程下，估计漂移系数和扩散系数中的有限维参数。假设采样发生在长度为$h_n$的规则间隔的时间间隔上，使得$n h_n=T$。渐近是当$h_n\to0$、$T\to\infty$和$n h_n^2\to 0$为$n\to\infty$时。证明了漂移系数和扩散系数参数估计的相合性和渐近正态性。
