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。社会科学和自然科学中的统计模型通常包括感兴趣的参数以及需要使用研究人员可用的数据估计的其他参数（干扰参数）。在这些模型中，所谓的半参数模型特别重要，因为它们灵活且对模型错误规范产生的偏差不太敏感。由于这些模型中的干扰参数是未知函数，研究人员在其估计中使用非参数技术，并且通常依靠渐近理论（假设大量数据的近似）进行统计推断。尽管半参数估计的经典渐近理论得到了很好的发展，但这些结果对于偏离所施加的基本假设通常是不稳健的。此外，半参数估计器的适用性通常受到其性能对估计过程中涉及的平滑和调谐参数的看似特别选择的敏感性的限制。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，研究是否有可能使用半参数估计器进行统计推断是至关重要的，这种估计器对非参数估计器的调谐和平滑参数选择的变化具有鲁棒性，并且对半参数模型的不可观察假设的偏离具有鲁棒性。该项目旨在为一类允许鲁棒统计推断的半参数估计提供非标准渐近理论。这个项目的主要焦点是一个特殊的，但重要的，半参数估计称为密度加权平均导数估计。对这个估计量获得的初步发现表明，我们提出的非标准渐近理论为统计程序的构建提供了基础，这些统计程序表现出某种形式的鲁棒性，从理论和经验的角度来看，这可能很有吸引力。本建议还讨论了该理论如何影响常用重采样过程的有效性，如何在应用中选择调优参数（同时与我们的非标准渐近一致），以及该思想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的推理，并使半参数推理对研究人员和决策者更具吸引力。

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.