Nonparametric Function Estimation

统计函数估计在科学研究的许多领域都非常重要。许多问题的核心是需要构建一条曲线来总结一组数据的某些重要方面，无论是直方图、生存曲线还是响应曲线。做好这一点对实验者来说至关重要，对管理者、决策者和各类政府官员来说也是如此。同样重要的是，所使用的方法和所提供的解决办法应易于解释和普遍有效，而不施加难以或甚至不可能在实践中验证的理论限制和/或假设。与此相关的还有对客观方法的需求，这种方法不太依赖于特定形式的模型，这也是很难证明的。这种客观性有多种形式，但自然导致研究所谓的函数估计的非参数方法。例如，最简单的例子之一可能是线性回归问题的研究。如何处理回归问题的两个重要方面，是随机误差分布的正态性要求和响应变量与测量协变量之间的关系是线性的假设。在某些设置中，已知不满足常态假设。在其他设置中，不能假设响应变量和测量协变量之间的关系是线性的，事实上，甚至不能假设是已知的。这个问题的非参数解决方案是构建一个连接响应变量和协变量的函数估计，该函数不假设任何特定的函数形式（如线性），并且对随机误差的分布尽可能少地做假设。***本建议的主旨是发展和更好地理解非参数方法，如上面描述的，以解决一系列不同的统计函数估计问题。我们研究一些现代问题，这些问题在大数据出现的情况下尤为重要。例如，我们考虑在称为稀疏渐近的上下文中对列联表的一些工作，其中问题的复杂性随着收集的数据的增加而增长。我们还研究了一类新的函数回归模型，其中函数扮演协变量的角色。这些模型允许人们在响应变量和某些种群特征的完整分布之间建立联系。最后，我们研究了在数据收集中使用所谓的基于秩的抽样设计对标准估计方法的影响。我们还开发了新的函数估计方法，专门适用于这些交替的抽样计划。

Statistical Function Estimation is extremely important in many areas of scientific research. At the heart of many problems is the need to construct a curve that summarizes some important aspect of a set of data, be it a histogram, a survival curve or a response curve. Doing this well is of utmost importance for experimenters, but also for administrators, decision makers and government officials of all kinds. It is often also important that the methods used and provided solutions be simple to interpret and generally valid, without imposing theoretical restrictions and/or assumptions that are difficult or even impossible to verify in practice. Linked to this is also the need for objective methodologies that are not too dependent on a specific form of model, again something that can be hard to justify. This objectivity takes many forms, but naturally leads to studying the so-called nonparametric methods of function estimation.***For instance, one of the simplest examples of this is perhaps the study of linear regression problems. Two important aspects of how regression problems are often approached, are the requirement of normality for the random error distribution and the assumption that the relationship between the response variable and the measured covariates is linear. In some setups, the assumption of normality is known not to be met. In other setups, the relationship between the response variable and the measured covariates cannot be assumed to be linear, and in fact, cannot even be assumed to be known. A nonparametric solution to this problem would be to construct an estimate of the function linking the response variable to the covariates that does not assume any specific functional form (like linearity) and makes as few assumptions as possible about the distribution of the random error.***The main thrust of this proposal is to develop and better understand nonparametric approaches, such as the one described above, to a range of different statistical function estimation problems. We study modern takes on some problems, which are especially relevant given the advent of big data. For instance, we consider some work on contingency tables in the context known as sparse asymptotics, where the complexity of problems grows as more data are collected. We also study a new family of functional regression models, where functions play the role of covariates. These models allow one to establish a link between a response variable and the full distribution of some population characteristics. Finally, we study the impact, on standard estimation methods, of using so-called rank-based sampling designs for data collection. We also develop new methods of function estimation specifically adapted to these alternate sampling plans.