Collaborative Research: Model Diagnostics in Regression and Tobit Regression Models with Measurement Errors

对预测者集合和响应之间的关系的统计建模通常通过回归分析来实现。在经典回归模型中，预报器和响应变量都被假定为直接可观测的。在测量误差回归模型中，不能直接观测到预报量，而是观测到了一些替代物。在Tobit回归模型中，只有当响应变量超过某个阈值时才能观察到它。在回归和Tobit回归模型中存在测量误差的情况下，开发有用的和最优的推断程序是理论和应用统计学中的主要问题。尽管有这种需要，但对有/无测量误差的测量误差回归模型和Tobit回归模型中的拟合度和不拟合度检验的研究一直滞后。在这个项目中，研究人员分析了变量误差和Berkson测量误差回归模型的随机分量分布的拟合优度检验，以及在有或没有这些测量误差的Tobit回归模型中回归函数的一些非参数估计。此外，研究人员开发并分析了具有这些测量误差的Tobit回归模型中的拟合缺失和拟合优度检验。研究人员在这些模型中提供了一些新的、有用的和最优的推理程序，并向统计学和相关学科的广泛专业受众深入了解了它们的理论属性。该项目处于模型检验的前沿，在回归和Tobit回归模型中的预测值存在测量误差的情况下。它推动和丰富了统计理论和方法，从而有助于填补统计学中存在的一个重大空白和公认的理论空白。测量误差在健康科学、物理科学、经济学和社会科学中非常普遍。例如，在研究饮食对乳腺癌的影响时，研究的预测乳腺癌的预测变量之一是无法精确测量的长期饱和脂肪摄入量。取而代之的是，在这种类型的调查中，经常使用对每个患者进行24小时饮食回忆的替代方法。同样，在研究辐射暴露对人类的影响时，一个人所暴露的确切辐射量经常被错误地测量。在劳动研究中，当研究女性的工作状态与她们的年龄、教育程度和工作经验等背景信息之间的关系时，测量误差的影响存在于教育变量(如母亲和父亲的教育经历)中。在这类研究中使用的Tobit回归模型往往存在测量误差问题。大多数涉及Tobit回归模型的实证研究都倾向于忽略测量误差，这往往会导致统计结论的偏颇和低效。本项目的研究重点是帮助评估回归模型或在存在测量误差的情况下随机分量分布模型的准确性，有助于为这些和其他类似例子开发更准确的统计推断。

Statistical modeling for relationships between a collection of predictors and a response is often implemented by regression analysis. In the classical regression model, both predictors and response variables are assumed to be directly observable. In measurement error regression models, predictors cannot be observed directly, instead, some surrogates are observed. In Tobit regression models, the response variable is observed only when it is above some threshold. The development of useful and optimal inference procedures in the presence of measurement errors in regression and Tobit regression models is of major concern in theoretical and applied statistics. Despite this need, the study of goodness-of-fit and lack-of-fit tests in the measurement error regression models and Tobit regression models with/without measurement errors has lagged behind. In this project, the investigators analyze goodness-of-fit tests for the distributions of the random components of errors-in-variables and Berkson measurement error regression models, and some nonparametric estimators of regression functions in Tobit regression models with or without these measurement errors. Furthermore, the investigators develop and analyze lack-of-fit and goodness-of-fit tests in Tobit regression models with these measurement errors. The investigators make available some new, useful, and optimal inference procedures in these models with an in-depth understanding of their theoretical properties to a wide professional audience in statistics and related disciplines. This project is at the cutting edge of model checking in the presence of measurement error in predictors in regression and Tobit regression models. It advances and enriches the statistical theory and methodology, thereby helping to fill a significant void and well recognized theoretical gap that exists in statistics. Measurement errors are very prevalent in the health sciences, physical sciences, economics, and the social sciences. For example, when investigating the effect of diet on breast cancer, one of the predictor variables studied for predicting breast cancer is the long-term saturated fat intake which cannot be measured precisely. Instead, the surrogate of a 24 hour diet recall for each patient is often used in this type of investigation. Similarly, the exact amount of radiation a person is exposed to when studying the effect of radiation exposure on humans is often measured with error. In labor studies, when investigating the relationship between women's working status and their background information, such as age, education and working experience, the effect of measurement errors is present in the education variables (such as mother's and father's education experience). Tobit regression models, which are used in these types of studies, often suffer from the measurement error problem. Most empirical studies involving Tobit regression models tend to ignore the measurement errors, which usually leads to biased and inefficient statistical conclusions. The research focus of this project, which helps in assessing the accuracy of a regression model or of a model for the distributions of random components in the presence of measurement errors, helps to develop more accurate statistical inference for these and other similar examples.