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

研究项目。回答从家庭决策到识别来自多种来源的信号的组成部分等问题需要对数据进行彻底的检查。通常，在统计分析中，简化假设（使清晰的答案成为可能）与认识到现实可能更为复杂之间存在矛盾。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“聚集”（失业资格截止点的劳动时间，或信号峰值）。我的研究项目是对不规则数据的非参数统计性质的理论评价。方法。统计性质通常是通过检验导数和展开式来确定的。在聚束的情况下，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”（Gel’fand, Shilov, 1964）来解决，有时也称为“分布”（L. Schwarz, 1964）。通过放弃测量距离的一些精度（“弱”拓扑），我们可以处理可微的广义函数。因此，我的建议通过考虑随机广义函数来检验统计的极限性质。过去的进步。在我的工作（2008年，2017年）中使用该方法来推导核密度估计器的极限过程，核密度估计器是核统计的构建块，例如用于回归函数。我和我的博士生（2014）推导了条件分布核估计量的性质和一个新的检验核估计量的统计量。在2014年的另外两篇论文中，我推导了卷积问题的解决方案，以将信号从噪声中分离出来。这说明了广义函数的实用性。预期的未来结果。我计划将重点放在三个目标上，在这些目标上我将应用广义函数。(1)发展了条件均值核估计和参数规范检验的极限过程，以处理具有聚类的数据分布。应用程序将为家庭决策（劳动力供给、服务需求）提供新的见解。(2)将所得逆问题的解应用于信号提取中盲源分解的新算法。(3)推导分布时间序列的极限性质。最近的研究结果（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。应用于经济、金融和自然科学的动态特征。HQP培训。我所从事的有前途的方法论为我指导的学生提供了获得复杂大数据模型非参数分析的前沿技能的机会。这样的分析对于实证研究是有价值的。

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.