Mathematical Sciences: Adaptive estimation of nonparametric curves

本研究探讨了在缺乏干扰参数先验信息的情况下，非参数曲线自适应估计的急剧风险收敛性。由于这种情况下的最优风险收敛关键取决于估计曲线的平滑性，因此目标是开发一个合适的估计器，而不需要先验的平滑性知识。具体的应用领域是概率密度估计、非参数回归、滤波和谱密度估计。从数据中对曲线的大多数（最优）估计依赖于对待估计曲线的平滑性的假设。自适应估计允许在收集观察（数据）时建立关于平滑度的信息。这是在估计曲线的过程中同时完成的。对于这种顺序方法，所需的数据量（样本量）取决于观测值和底层曲线。这两种理论方法的最优总体策略和一些应用将被考虑。

This research examines sharp risk convergence for adaptive estimates of nonparametric curves in the absence of prior information on nuisance parameters.Since optimal risk convergence in this context depends crucially on the smoothness of the estimated curve, the goal is to develop a suitable estimator without a priori knowledge of the smoothness. Particular areas of application are probability density estimation, nonparametric regression, filtering and spectral density estimation. Most (optimal) estimates of curves from data rely on assumptions about the smoothness of the curve to be estimated. Adaptive estimation allows information about the smoothness to be built up as observations (data) are gathered. This is done at the same time and as part of the process of estimating the curve itself. For sequential methods of this kind, the amount of data required (sample size) depends on the observations and the underlying curve. Both theoretical approaches to an optimal overall strategy and some applications will be considered.