Adaptive estimation of mixed discrete-continuous distributions under smoothness and sparsity

经济学家经常分析这样的问题，比如消费者是否会买车（是或否），如果会的话，在车上花多少钱（不计价值的一笔钱）。经济学家没有一个好的方法来分析这些问题。该研究项目将开发一种新的有效方法来分析具有这些特征的数据。这一结果将为社会科学研究人员分析涉及复杂数据结构的问题提供一个重要工具。随着越来越复杂的数据收集和强大的计算能力的可用性，一种能够有效地分析这些复杂数据集的方法对各个领域的研究人员和决策者来说都将是非常有价值的。在这项研究中开发的方法将被编程并免费提供给所有研究人员。这项研究的结果将使研究人员能够向决策者提供更准确的建议，从而改善决策和经济增长。本研究项目开发了一个估计混合离散-连续分布的框架，其中分布的多元离散部分可以有大量或少量的支撑点；可能是光滑的，也可能不是，这些特征可以从一个离散坐标到另一个离散坐标。在这些情况下，将推导出离散-连续分布估计的最优收敛率。初步结果表明，平滑只对具有快速增长的支撑点数量或足够高的平滑水平的离散变量子集有益。所提出的估计程序是基于贝叶斯混合的多元正态分布与协变量相关的混合权。所提出的方法将为标准计量模型（如有序probit和泊松回归）和结构离散选择模型的两阶段估计程序中的第一阶段估计器提供实用和最优的自适应非参数替代方案。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Economists often analyze questions such as whether a consumer will buy a car or not (yes or no) and if so how much to spend on the car (an amount that takes any value). Economists do not have a good method to analyze these questions. This research project will develop a new and efficient method to analyze data that has these characteristics. The results will give social science researchers an important tool to analyze problems that involve complicated data structures. With ever increasing sophisticated data collection and the availability of powerful computing power, a method that can be used to efficiently analyze such complicated data sets will be extremely valuable to researchers and policy makers in all fields. The methods developed in this research will be programmed and be freely available to all researchers. The results of this research will allow researchers to give more accurate advice to policy makers and thus improve decision making and economic growth. This research project develops a framework for estimating mixed discrete-continuous distributions where the multivariate discrete part of the distribution can have either a large or a small number of support points; may be smooth or not, and these characteristics can differ from one discrete coordinate to another. The optimal convergence rates for estimation of discrete-continuous distributions will be derived in these settings. Preliminary results suggest that smoothing is beneficial only for a subset of discrete variables with a quickly growing number of support points or sufficiently high level of smoothness. The proposed estimation procedures are based on Bayesian mixtures of multivariate normal distributions with covariate dependent mixing weights. The proposed methods will deliver practical and optimal adaptive nonparametric alternatives to standard econometric models such as ordered probit and Poisson regression, and first stage estimators in two stage estimation procedures for structural discrete choice models.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.