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

点击翻译按钮获取中文摘要

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.