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.