Mathematical Sciences: A Study of Nonparametric Curve Estimators

这项研究的一个主要目的是发展一种统一的理论，用于基于光滑样条函数和核估计的似然函数对多变量响应曲面进行非参数拟合。该方法将惩罚似然和加权似然的概念结合在一个惩罚加权的负对数似然方程中。这将提供一个共同的框架，在其中研究和比较核估计器和平滑样条线的性质。剩下的目标是解决如何使核估计器尽可能有效，以证明非参数替代目前流行的参数方法中使用的响应面方法。这项研究属于统计回归的一般领域。该领域最近的许多活动都是为了放松回归理论中的假设，从而增加这些统计方法的实用性。最近进展的结果是存在一套新的零散的数据分析工具，它们彼此之间的利弊不清楚。这阻碍了它们在其他科学问题上的应用。这项提议是为了统一这一领域的最新进展，从而为选择最适当和信息最丰富的工具来分析特定的数据集提供手段。

A primary objective of this research is to develop a unified theory for nonparametric fitting of multivaritate response surfaces based on likelihoods for smoothing splines and kernel estimators. The proposed approach involves combining the notions of a penalized likihood and a weighted likelihood in a penalized- weighted negative log-likelihood equation. This will provide a common framework within which to study and compare the properties of kernel estimators and smoothing splines. Remaining objectives address ways with which to make kernel estimators as efficient as possible in order to prove a nonparametric alternative to the currently popular parametric approaches used in response surface methodoloy. This research is in the general area of statistical regression. Much recent activity in the field has been directed at relaxing the assumptions in regression theory and thereby increasing the utility of these statistical methods. The result of recent progress is the existance a new set of fragmented tools for data analysis with unclear advantanges and disadvantages with respect to each other. This inhibits their application to other scientific problems. This proposal is to unify recent advances in this area and, thereby, provide the means to choose the most appropriate and informative tool for analyzing a particular data set.