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

非参数识别、估计和推理：广义函数方法

• 批准号: RGPIN-2020-05444
• 负责人: ZindeWalsh, Victoria
• 金额: $ 1.31万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=740199
• 关键词: Non; parametric; identification; estimation; inference

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.

研究计划。回答从家庭决策到识别来自混合来源的信号组成部分等问题需要对数据进行彻底检查。通常，在统计分析中，简化假设（使清晰的答案成为可能）与现实可能更加复杂的认识之间存在着紧张关系。非参数统计处理数据的一般分布和关系形式。它们在应用中取得了成功，可以测试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”（失业资格截止点的工时，或信号峰值）。我的研究项目是对不规则数据的非参数统计特性的理论评估。方法论。 统计特性通常是通过检查导数和展开式来建立的。通过聚束，导数不再像普通函数那样存在。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）。通过放弃一些测量距离的精度（“弱”拓扑），我们可以使用可微分的广义函数。因此，我的建议通过考虑随机广义函数来检查统计的极限属性。 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器的极限过程，该估计器是核统计的构建块，例如为回归函数。我和我的博士生（2014）推导出了条件分布核估计量的属性以及用于测试它的新统计量。在 2014 年的另外两篇论文中，我导出了卷积问题的解决方案，以将信号与噪声分开。这显示了广义函数的有用性。预期的未来结果。我计划重点关注三个目标，并在其中应用通用函数。 (1) 开发条件均值的核估计器和参数规范测试的极限过程，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)应用我工作中导出的反问题的解来构建信号提取中盲源分解的新算法。 (3) 推导时间序列分布的极限性质。最近的结果（Chang 等，2016）使用大数据进行密度随机过程；我会考虑一般分布。应用到经济、金融和自然科学的动态特征。 总部培训。我正在研究的有前途的方法为我指导下的学生提供了获得复杂大数据模型非参数分析尖端技能的机会。这种分析对于实证研究很有价值。