Quantile regression with mismeasured or missing covariates

该奖项是根据《美国复苏和再投资法案》(2009年第111-5号公法)资助的)。分位数回归(Koenker and Bassett，1978)已成为一种重要的统计方法，并被广泛应用于经济学、生物学、生态学和金融学等领域。通常情况下，数据集并不是完全获得的。一些变量的测量可能会有误差，而其他变量可能包含遗漏的观测值。忽视测量误差或遗漏观测值可能会导致估计中的严重偏差。因此，如何处理测量误差和缺失数据已经产生了大量的文献。遗憾的是，现有的大多数方法依赖于参数似然形式，因此不能直接应用于分位数回归。这项建议的目标是开发方法和理论，以便即使在存在测量误差和/或遗漏观测的情况下也能获得无偏分位数估计。本项目下拟开展的具体研究活动包括以下四个方面。(1)发展了允许测量误差存在的线性分位数模型的估计方法，并研究了由此得到的估计量的渐近性质。(2)将线性分位数模型的估计方法推广到半参数模型，具有更大的灵活性，便于更广泛的应用。(3)开发相关推理和模型充分性评估工具。(4)对1-3中提出的方法进行了扩展，以解决条件分位数模型中的缺失数据问题，包括估计、推理和模型评估。统计方法包括分位数回归、测量误差和缺失数据问题的方法和理论、非参数和半参数建模、拟合度检验、自举方法和稳健统计。由于流行病学、艾滋病研究、遗传学、癌症研究和环境科学等领域中普遍存在测量误差和缺失数据，因此本研究将在这些应用中提供更准确的推断和更全面的资格。调查人员将制定的方法对统计研究具有普遍意义。拟议的研究将通过出版物、在国内和国际会议上的陈述以及与临床和公共卫生研究人员的合作来广泛传播。

This award is funded under the American Recovery and Reinvestment Act of 2009 Public Law 111-5). Quantile regression (Koenker and Bassett, 1978) has emerged as an important statistical methodology, and has been used in a wide range of applications including economics, biology, ecology and finance. Very often a data set is not perfectly obtained. Some variables may be measured with error, while others may contain missing observations. Ignoring measurement errors or missing observations could lead to substantial bias in estimation. For this reason, how to handle measurement errors and missing data has generated a large number of literatures. Unfortunately, most of the existing methods rely on a parametric likelihood form, and hence cannot be applied to quantile regression directly. This proposal targets at developing methods and theories for obtaining unbiased quantile estimates even in the presence of measurement errors and/or missing observations. The specific proposed research activities under this project include the following four aspects. (1) Develop estimation methods for linear quantile models allowing the existence of measurement errors, and investigate the asymptotic properties for the resulting estimator. (2) Extend the estimation method for linear quantile model to semiparametric models, which brings more flexibility and hence facilities a wider range of applications. (3) Develop related inference and model adequacy assessment tools. (4) Extend the proposed methods in 1 - 3 to address missing data problems in conditional quantile models, including estimation, inference and model assessment. The statistical methods to be employed for this proposal cover quantile regression, methods and theories for measurement errors and missing data problems, nonparametric and semi-parametric modeling, goodness-of-fit tests, bootstrapping methods and robust statistics.The proposed research will lead to more accurate inference and more comprehensive qualifications in various research applications in epidemiology, HIV research, genetics, cancer research and environmental science, as measurement errors and missing data commonly exist in those applications. The methodologies to be developed by the investigators are of general interest to statistical research. The proposed research will be widely disseminated through publications, presentations in domestic and international conferences, and collaborations with clinical and public health researchers.