General Semiparametric Inference via Bootstrap Sampling

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The research objectives of this project are first to prove the theoretical validity of the bootstrap method as a general inferential tool for the semiparametric models, and then invent a computationally attractive bootstrap inference procedure, called k-step bootstrap. Semiparametric modelling has provided an excellent framework for the modern complex data due to its flexibility to model some features of the data parametrically but without assuming anything for the other features. The bootstrap is the most popular data-resampling method used in statistical analysis, and has recently been applied to the semiparametric models arising from a wide variety of contexts. Therefore, the systematic theoretical studies on the bootstrap inferences for the semiparametric models are fundamentally important. In practice, the computational cost of the bootstrap inference procedure is particularly high for the semiparametric models. Thus, the investigator proposes an approximate bootstrap method, i.e. k-step bootstrap, and will show that this novel approach results in huge computational savings but without sacrificing any degree of inference accuracy. In addition, the investigator will develop a set of asymptotic results to elucidate the asymptotic structure of the semiparametric M-estimation, which is crucial for the future theoretical research. M-estimation refers to a general method of estimation including the maximum likelihood estimation as a special case.The primary impact of the proposed work is to lay solid theoretical foundation for the general semiparametric inferences via bootstrap sampling. In addition, the proposed k-step bootstrap approach is practically beneficial in several regards. For instance, the scientists who bootstrap a large data set will benefit, as the minimal computational cost needed in the k-step bootstrap to achieve the satisfactory inference accuracy will be precisely analyzed. However, the broader impacts of the proposed activities are multiple. For instance, a key aspect of this project is the integration of research and teaching, which will be achieved by proposing specific projects for students during the teaching of classes on semiparametric inferences and bootstrap computation. This pedagogical method also facilitates the participation of underrepresented groups of students.