Collaborative Research: Small Bandwidth Asymptotic Theory for Kernel-Based Semiparametric Estimators

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。社会和自然科学中的统计模型通常包括一个感兴趣的参数以及其他需要使用研究人员可用的数据进行估计的参数（滋扰参数）。在这些模型中，所谓的半参数模型是特别重要的，因为它们是灵活的，不太敏感的模型误设定所产生的偏见。由于这些模型中的干扰参数是未知函数，研究人员在估计中使用非参数技术，并且通常依赖于渐近理论（假设大量数据的近似）进行统计推断。虽然经典的半参数估计的渐近理论发展得很好，这些结果一般是不稳健的偏离所施加的基本假设。此外，半参数估计的适用性往往是有限的敏感性，他们的表现似乎特设的选择平滑和调整参数的估计过程中涉及。这种稳健性的缺乏通常会导致不正确的统计推断，这可能会导致研究人员和政策制定者从使用这些半参数估计的实证工作中得出有缺陷的结论。因此，研究是否可能使用半参数估计进行统计推断是至关重要的，这种半参数估计对非参数估计的调整和平滑参数选择的变化具有稳健性，以及偏离半参数模型的不可观察假设。这个项目旨在为一类半参数估计提供非标准的渐近理论，允许稳健的统计推断。这个项目的主要重点是一个特殊的，但重要的，半参数估计称为密度加权平均导数估计。初步研究结果表明，我们提出的非标准渐近理论提供了建设的统计程序，表现出一定形式的鲁棒性，可能是吸引力从理论和经验的角度来看的基础。这个建议还讨论了这个理论如何影响常用的reservation程序的有效性，如何调整参数可能会选择在应用程序中（同时符合我们的非标准渐近），以及这个想法是否可以更广泛地应用于其他半参数估计。这项研究的结果预计将有利于几个研究领域，从经济学或政治学到生物统计学或公共卫生，使研究人员能够在半参数模型中进行稳健的推断，并使半参数推断对研究人员和政策制定者更具吸引力。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).Statistical models in social and natural sciences typically include a parameter of interest as well as other parameters that need to be estimated (nuisance parameters) using the data available to the researcher. Among these models, the so-called semiparametric models are of particular importance because they are flexible and less sensitive to biases generated by model misspecification. Because the nuisance parameters in these models are unknown functions, researchers use nonparametric techniques in their estimation and usually rely on asymptotic theory (approximations that assume a large amount of data) to conduct statistical inference. Although classical asymptotic theory for semiparametric estimators is well developed, these results are in general not robust to departures from the underlying assumptions imposed. Moreover, the applicability of semiparametric estimators is often limited by the sensitivity of their performance to seemingly ad hoc choices of smoothing and tuning parameters involved in the estimation procedure. This lack of robustness usually translates in incorrect statistical inference that may lead researchers and policy-makers to draw flawed conclusions from empirical work that employs these semiparametric estimators.As a consequence, it is crucial to investigate whether it is possible to conduct statistical inference using semiparametric estimators that is robust to changes in the tuning and smoothing parameters choices underlying the nonparametric estimator, and to departures from the unobservable assumptions underlying the semiparametric model. This project seeks to provide non-standard asymptotic theory for a class of semiparametric estimators that allows for robust statistical inference. The main focus of this project is on a particular, yet important, semiparametric estimator called the density-weighted average derivative estimator. Preliminary findings obtained for this estimator, show that our proposed non-standard asymptotic theory provides the basis for the construction of statistical procedures that exhibit certain forms of robustness that may be appealing from both theoretical and empirical perspectives. This proposal also discusses how this theory affects the validity of commonly used resampling procedures, how tuning parameters may be selected in applications (while being consistent with our non-standard asymptotics), and whether this idea may be applied more broadly to other semiparametric estimators. 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 policy-makers.