Non-parametric identification, estimation and inference: generalized functions approach

研究计划。从家庭决策到识别来自混合来源的信号的组成部分，这些问题都需要对数据进行彻底的检查。通常在统计分析中，简化假设使尖锐的答案成为可能，而现实可能更加复杂，这两者之间存在着张力。非参数统计处理数据的一般分布和关系的形式。它们在实际应用中是成功的，可以检验尖锐参数模型的有效性。然而，广泛使用的方法通常依赖于对数据的假设，例如排除“聚束”（失业资格截止时的劳动时间或信号峰值）。我的研究项目是非参数统计与不规则数据的性质的理论评估。方法论 统计性质通常通过检验导数和展开式来建立。对于聚束，导数不作为普通函数存在。幸运的是，缺乏可微性的问题可以通过“广义函数”（Gelfand，Shilov，1964）来解决，有时也称为“分布”（L。施瓦茨，1964年）。通过放弃一些测量距离的精度（“弱”拓扑），我们可以处理可微的广义函数。因此，我的建议审查的极限性质的统计考虑随机广义函数。 过去的进展。该方法在我的工作（2008，2017）中用于推导核密度估计的极限过程，这是核统计的构建模块，例如回归函数。我和我的博士生（2014）推导出了条件分布的核估计量的性质和一个新的测试统计量。在另外两篇2014年的论文中，我推导出了卷积问题的解决方案，以将信号从噪声中分离出来。这表明了广义函数的有用性。预期未来成果。我计划集中在三个目标，我将应用广义函数。(1)为条件均值的核估计和参数规格的检验开发极限过程，以处理具有聚束的数据分布。应用程序将为家庭决策（劳动力供应，服务需求）提供新的见解。 (2)应用本文所导出的反问题的解，构造了一种新的盲源分解信号提取算法。(3)推导分布时间序列的极限性质。最近的结果（Chang et al，2016）使用大数据进行密度的随机过程;我将考虑一般分布。应用是经济学、金融学和自然科学中的动态特征。 HQP的培训。我正在研究的有前途的方法为我指导下的学生提供了机会，使他们能够获得对复杂和大数据模型进行非参数分析的尖端技能。这种分析对于实证研究具有一定的参考价值。

Research program. Answering questions ranging from household decisions to identifying components of a signal coming from a mix of sources requires a thorough examination of data. Typically in statistical analysis there is a tension between simplifying assumptions that make sharp answers possible and the realization that reality may be more complicated. Non-parametric statistics tackle general distributions of data and forms of relations. They are successful in applications and can test validity of sharp parametric models. However, widely used methods often rely on assumptions about the data that e.g. exclude "bunching" (labor hours at the cut-off for unemployment eligibility, or spike in signal). My research program is theoretical evaluation of the properties of non-parametric statistics with irregular data. Methodology. Statistical properties are usually established by examining derivatives and expansions. With bunching the derivatives do not exist as ordinary functions. Fortunately, the problem of lack of differentiability can be solved by "generalized functions" (Gel'fand, Shilov, 1964), sometimes called "distributions" (L. Schwarz, 1964). By giving up some precision in measuring distances ("weak" topology) we can work with generalized functions that are differentiable. Thus my proposal examines the limit properties of statistics by considering random generalized functions. Past progress. The methodology was used in my work (2008, 2017) to derive the limit process of the kernel density estimator which is the building block for kernel statistics, e.g. for regression function. My PhD student and I (2014) derived the properties for kernel estimator of conditional distribution and a new statistic for testing it. In two other 2014 papers I derived solutions to convolution problems to disentangle the signal from noise. This showed usefulness of generalized functions. Expected future results. I plan to focus on three objectives where I will apply generalized functions. (1) Developing the limit process for the kernel estimator of conditional mean and tests of parametric specifications, to work with data distributions with bunching. Applications will provide new insights for household decisions (labor supply, demand for services). (2) Applying the solutions to inverse problems derived in my work to construct a new algorithm for blind source decomposition in signal extraction. (3) Deriving limit properties for time series of distributions. There are recent results (Chang et al, 2016) that use big data for stochastic processes of densities; I will consider general distributions. Applications are to dynamic features in economics, finance and natural sciences. Training of HQP. The promising methodology that I am working on provides opportunities for students under my direction to acquire cutting-edge skills for non-parametric analysis of models with complicated and big data. Such analysis is valuable for empirical research.