Collaborative Research: Non-Standard Asymptotic Theory for Semiparametric Estimators

合作研究：半参数估计的非标准渐近理论

• 批准号: 1122994
• 负责人: Matias Cattaneo
• 金额: $ 28.39万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1122994&HistoricalAwards=false
• 关键词: Collaborative; Research; Non; Standard; Asymptotic

Modern statistical and econometric models in social and natural sciences are complex and typically include many unknown parameters. Some of these parameters are of particular interest to the researchers and policymakers (e.g., the mean effect of a treatment), while others are not (e.g., the exact form of a regression function or the probability law of the observed covariates). The latter parameters are usually called nuisance parameters because their values are needed in order to conduct valid statistical inference on the parameters of interest, even though the researchers are not interested in them. An important class of these models are the so-called semiparametric models, which have the distinctive feature that the nuisance parameters are functions (as opposed to numbers). A modern, well-established approach in statistics and econometrics to conduct inference in semiparametric models is to estimate in a flexible, non-parametric way the nuisance parameters first (using the available data), and then employ these estimates as a preliminary guess of their true value in order to conduct inference on the parameters of interest. This procedure is generically referred to as semiparametric inference, and is particularly useful because of its flexibility and lack of sensitivity to biases generated by model misspecification.Semiparametric inference procedures are very popular among theoretical researchers, partially because of their nice and well understood large sample properties (approximations that assume a large amount of data). However, these inference procedures are considerably less popular among empirical researchers and policymakers, mainly because they are known to be highly sensitive to the way that they are implemented in practice. Specifically, an important drawback of most semiparametric inference procedures is that they rely on non-parametric techniques for the estimation of the nuisance parameters, which in turn require the selection of tuning and smoothing parameters. These additional parameters are artificially introduced in the inference procedure to flexibly approximate the unknown functions (the nuisance parameters). The large sample approximations employed in the literature ignore the effect of these additional parameters that are artificially introduced in the construction of the inference procedure. This fact, in turn, leads to an important lack of robustness of semiparametric inference procedures, that is, small changes in the choice of tuning and smoothing parameters lead to dramatically different empirical results, making applied work unreliable in general. In other words, this lack of robustness usually translates in incorrect statistical inference that may lead researchers and policymakers to draw flawed conclusions from empirical work that employs these semiparametric inference procedures.The main goal of the proposed research agenda is to develop new, alternative large sample approximations to commonly used semiparametric inference procedures that (at least partially) account for the effect of the specific user-defined choices of tuning and smoothing parameters involved in the inference procedure. This alternative asymptotic theory leads to more "robust" statistical inference procedures because it captures the effect of certain terms that are assumed away by the conventional large sample approximations. This project will proceed in two main stages. First, alternative large sample approximations will be developed for specific semiparametric examples, including weighted averaged derivatives and partially linear model. Not only these models are of interest in their own right, but also they will provide some of the key ingredients to understand the new theoretical features emerging from the non-standard large sample approximations studied in this proposal. Among other problems, the goal is to establish an alternative first-order large sample distribution, derive valid standard-error estimators, develop new ways of selecting the value of the tuning and smoothing parameters, study the validity of commonly used resampling procedures, and explore the higher-order implications of the alternative asymptotic approximations. Once the study of these particular semiparametric procedures is well understood, the second stage of the investigation will be to develop a generalization and unification of the theoretical results outlined for the special examples, which will cover many other problems of interest.The results of this research are expected to benefit several fields of study, ranging from Economics or Political Science to Biostatistics or Public Health, allowing researchers to conduct "robust" inference in semiparametric models, and making semiparametric inference more attractive to researchers and policymakers. To further increase the impact of this research proposal, a key goal is to provide computer code for commonly used platforms, and to write a non-technical survey with a discussion on theory and implementation of both the classical results and the new results emerging from the research proposed.

社会科学和自然科学的现代统计和计量经济模型是复杂的，通常包括许多未知参数。其中一些参数是研究人员和决策者特别感兴趣的（例如，治疗的平均效果），而其他参数则不是（例如，回归函数的确切形式或观察到的协变量的概率律）。后一种参数通常被称为干扰参数，因为即使研究人员对它们不感兴趣，也需要它们的值来对感兴趣的参数进行有效的统计推断。这些模型中有一类重要的是所谓的半参数模型，它有一个独特的特征，即讨厌的参数是函数（而不是数字）。在统计和计量经济学中，在半参数模型中进行推理的一种现代的、成熟的方法是以一种灵活的、非参数的方式首先（使用可用的数据）估计干扰参数，然后将这些估计作为对其真实值的初步猜测，以便对感兴趣的参数进行推理。这个过程通常被称为半参数推理，由于其灵活性和对模型错误规范产生的偏差缺乏敏感性而特别有用。半参数推理过程在理论研究人员中非常流行，部分原因是它们具有良好且易于理解的大样本特性（假设大量数据的近似）。然而，这些推理程序在实证研究人员和政策制定者中相当不受欢迎，主要是因为它们对实践中实施的方式高度敏感。具体来说，大多数半参数推理过程的一个重要缺点是它们依赖于非参数技术来估计干扰参数，这反过来又需要选择调谐和平滑参数。在推理过程中人为地引入这些附加参数，以灵活地逼近未知函数（干扰参数）。文献中采用的大样本近似忽略了这些额外参数的影响，这些参数是在推理过程的构建中人为引入的。这一事实反过来又导致半参数推理过程缺乏鲁棒性，也就是说，调整和平滑参数选择的微小变化会导致显著不同的经验结果，使应用工作总体上不可靠。换句话说，这种鲁棒性的缺乏通常转化为不正确的统计推断，这可能导致研究人员和政策制定者从采用这些半参数推理程序的实证工作中得出有缺陷的结论。提出的研究议程的主要目标是为常用的半参数推理过程开发新的、可替代的大样本近似值（至少部分地）解释特定用户自定义选择的调谐和平滑参数在推理过程中所涉及的影响。这种可选的渐近理论导致了更“稳健”的统计推断过程，因为它捕获了传统的大样本近似所假定的某些项的影响。这个项目将分两个主要阶段进行。首先，将为特定的半参数示例开发替代的大样本近似，包括加权平均导数和部分线性模型。这些模型不仅本身就很有趣，而且它们将提供一些关键成分，以理解本提案中研究的非标准大样本近似中出现的新理论特征。在其他问题中，目标是建立一个可选的一阶大样本分布，推导有效的标准误差估计量，开发选择调谐和平滑参数值的新方法，研究常用重采样程序的有效性，并探索可选渐近近似的高阶含义。一旦对这些特殊的半参数过程的研究被很好地理解，调查的第二阶段将是发展对特殊例子概述的理论结果的概括和统一，这将涵盖许多其他感兴趣的问题。这项研究的结果有望使几个研究领域受益，从经济学或政治学到生物统计学或公共卫生，允许研究人员在半参数模型中进行“稳健”推理，并使半参数推理对研究人员和政策制定者更具吸引力。为了进一步增加本研究计划的影响，一个关键目标是为常用平台提供计算机代码，并撰写一份非技术调查，讨论经典结果和新结果的理论和实现。