Curve Estimation Involving Time Series

DMS 9625412 Efrom movich这项关于曲线估计的研究集中于最优自适应非参数时间序列估计器，它具有：(I)对不同的损失函数渐近有效，(Ii)与基于潜在曲线的同类预言相比，在样本量较小的情况下表现良好，(Iii)对于水质监测或通过海洋磁异常分析全球变化等环境问题产生的数据压缩有效；(Iv)对噪声分布和长协方差观测具有鲁棒性。渐近分析是基于对局部经验过程、混合序列的现代概率结果和尖锐的数据驱动的谱密度估计的研究。从理论和数值两个方面探讨了小样本情况下的数据驱动估计，并通过密集的蒙特卡罗研究对数据驱动估计进行了研究。这项研究涉及为恢复不同科学问题中出现的图像开发最佳方法，这些问题包括：(I)激素的分泌，如胰岛素分泌，由于有噪音、长期依赖和间接的数据，迄今没有其他方法成功；(Ii)监测水质，其基础是对经过高效、密集计算的数据压缩的大型、相关和稀疏数据集的分析；(Iii)地质事件的时间，包括板块运动和气候变化；(Iv)新材料的测试和建模特性。***

DMS 9625412 Efromovich This research on curve estimation focuses on optimal adaptive nonparametric time series estimators that are: (i) asymptotically efficient for different loss functions, (ii) well performing for the case of small sample sizes in comparison with peer oracles based on underlying curve, (iii) efficiently data-compressing for problems arising in environmental problems like monitoring quality of water or the analysis of global change via marine magnetic anomaly; (iv) robust to distribution of noise and long covariance observations. The asymptotic analysis is based on the study of local empirical processes, modern probabilistic results for mixing sequences and sharp data-driven estimators of spectral density. Data-driven estimators for the case of small sample sizes are explored both theoretically via oracle inequalities and numerically via intensive Monte Carlo study. %%% The research involves the development of optimal methods for the recovery of images that arise in different scientific problems including: (i) secretion of hormones such as insulin secretion where no other methods have been successful so far due to noisy, long-dependent and indirect data; (ii) monitoring quality of water based on the analysis of large, correlated and sparse data sets that have undergone efficient, computationally intensive data-compression; (iii) timing of geological events, including plate motions and climatological variations; (iv) testing and modeling properties of new materials. ***