Nonparametric Function Estimation

点击翻译按钮获取中文摘要

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.