Robust Inferences Based on Minimum Distance for Semiparametric Models

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Statistical inference necessarily is based on statistical models for data. During most of the history of the subject, these have been parametric: the mechanism generating the data could be identified by specifying a few real parameters. During the last thirty years nonparametric and semiparametric models have flourished. Nonparametric model leaves the underlying distribution of data essentially free, while semiparametric model is a model between parametric and nonparametric. For semiparametric models, many authors have considered efficient and adaptive estimation while the robustness has been paid little attention. In most situations where a statistical model is assumed to model data, the model is only an approximation to reality and the data could be noisy and subject to outliers. When outliers are present, classical methods such as maximum likelihood estimation will be distorted away from the true parameter values. Therefore, there is a strong demand of robust statistical inferences that always work well within neighborhoods of the putative model and are not unduly influenced by model misspecification and outlying observations. Following this direction, this proposed research aims to develop robust inferences for semiparametric models. For this purpose, I propose to use minimum distance (MD), particularly minimum Hellinger distance (MHD), approach that has been barely attempted for semiparametric models. Compared with parametric models, the additional unknown nonparametric component in semiparametric models adds an extra degree of complexity and difficulty to the construction of MD based inferences and to the study of corresponding asymptotic properties. This proposed research includes four parts: A. Robustness and efficiency study of MD estimation for general semiparametric model; B. Construction of robust efficient estimation based on MD for semiparametric regression models; C. Construction of robust efficient estimation based on MD for semiparametric mixture models; D. Applications of MD based robust inferences in genetic studies. These four parts encompass fundamental development in theorem, concrete implementation for specific models, and practical application to real-world problems.