Mathematical Sciences: Mathematical Statistics on Singular Problems

奇异函数估计问题覆盖了统计理论的很大一部分。一些具有最优速率的程序已经被很好地建立起来；但最近关于白噪声问题的工作已经导致了基于泛函模数的统一方法。对于许多重要的统计问题，这种方法非常简单地给出了最优或接近最优的程序。对于一些统计估计问题，最优方法是众所周知的，但对于更多的人来说，要么是方法的最优性不确定，要么是它的效率未知。这项研究将产生普遍有效的程序，可以有许多科学应用，例如复杂的问题，如从断层扫描或其他成像技术重建“真实图像”。这种统一的估计方法应该在工程、生物、经济和其他领域得到应用，在这些领域，实际和理论科学问题都是用统计模型来框架的。

Singular functional estimation problems cover an extremely large portion of statistical theory. Some procedures with optimal rates have been well established; but recent work on white noise problems has led to a unified approach based on the modulus of the functional. This approach gives optimal or near optimal procedures quite simply for many important statistical problems. For some statistical estimation problems, optimal methods are well known, but for many more either optimality of the methodology is uncertain or its efficiency is unknown. This research will produce generally efficient procedures which can have many scientific applications, for example complex problems like reconstruction of the "true image" from tomography or other imaging techniques. This unified approach to estimation should find applications in engineering, biology, economics and other fields where practical and theoretical scientific problems are framed in terms of statistical models.