Functional linear regression

信息处理技术的创新和存储能力的提高使经济、金融、气象和医学等各个领域的连续数据收集成为可能。研究人员、公司和政府都在寻找利用这些丰富信息的方法。我的长期目标是开发统计工具来处理这一不断增长的数据量，并确保所有信息都得到利用。将这些数据视为曲线是解决这个问题的一种优雅方式。*本研究议程侧重于回归，其中回归变量和响应都是函数(或曲线)。虽然有许多关于输出为实数的函数线性回归的文章，但考虑函数输出的文章很少。我计划做出各种贡献。首先，我将提出一种新的基于Tikhonov正则化的估计量。该估计器将仅依赖于一个调谐参数，即正则化参数。第二，将制定一项规范测试。第三，我将讨论离散数据的问题。第四，将模型推广到回归变量为内生变量但存在其他外生变量的情况。我将发展一种新的估计量的理论，它可以被认为是工具变量估计量的推广。最后，将考虑延长时间序列。*一项广泛的模拟研究将显示估计器在实践中的表现如何。我计划将该模型应用于安大略省的电力市场。我有权获得安大略省所有工厂(包括水电、火力发电站和风力涡轮机)的电力需求和产量的数据集。我计划将这些数据与天气数据进行匹配，以建立一个电力需求模型。由于电力生产是内生的，温度可以用作仪器。这个模型将让我测试一些关于植物战略行为的重要经济学问题。*我计划开发的方法，适用于任何非常频繁地观察到时间数据或多个位置的空间数据的地区。这项研究的结果将有助于理论研究人员和应用统计学家获得更可靠的估计，并做出更好的决策。此外，我将为我的方法提供计算机代码，以确保更快地传递知识，并使其更容易应用。**

The technological innovations in information processing and the increased storage capability have made possible to collect continuous-time data in various fields such as economics, finance, meteorology and medicine. Researchers, companies, and governments look for ways to exploit this rich information. My long term objective is to develop statistical tools to deal with this ever growing amount of data and to make sure that all the information is taken advantage of. Looking at these data as curves is an elegant way to address this issue. ***This research agenda focuses on regressions where both the regressor and the response are functions (or curves). While there are many papers dealing with functional linear regression where the output is real, there are only a few considering a functional output. I plan to make various contributions. First, I will propose a new estimator based on Tikhonov regularization. This estimator will depend on only one tuning parameter, the regularization parameter. Second, a specification test will be developed. Third, I will address the issue of discrete data. Fourth, the model will be extended to the case where the regressor is endogenous but other exogenous variables are available. I will develop the theory for a new estimator which can be thought of as a generalization of the instrumental variable estimator. Finally, a time-series extension will be considered. ***An extensive simulation study will show how well the estimators perform in practice. I plan to apply the model to the electricity market in Ontario. I have access to a data set of the electricity demand and production from all plants in Ontario (including hydroelectricity, thermal power station, and wind turbines). I plan to match these data with weather data to build a model of electricity demand. As the electricity production is endogenous, the temperature can be used as instrument. This model will allow me to test some important economic questions about the strategic behavior of plants. ***The methods, I plan to develop, apply to any area where temporal data are observed very frequently or spatial data are observed for multiple locations. The results derived in this research will help both theoretical researchers and applied statisticians to obtain more reliable estimates and make better decisions. Moreover, I will provide computer codes for my methods to insure a faster transmission of knowledge and make it easier to apply.**