Variable Selection and Related Problems

建议的研究将发展和评估多元回归和相关问题的变量选择程序。贝叶斯和频率论的观点都将被考虑。从贝叶斯的角度来看，重点将放在随机搜索变量选择方法上，该方法使用层次混合模型来指导变量选择。这些方法将包括一次处理数百个变量的快速贝叶斯变量选择程序，广义线性模型中的变量选择程序，跨可交换回归的变量选择程序，以及同时执行变量选择和异常值去除的程序。从频率主义者的角度来看，重点将放在评估选择标准的新标准上，即风险膨胀。本研究将需要将风险膨胀的应用和推广到广泛的模型选择问题，包括广义线性模型，时间序列模型的顺序选择程序，以及变点问题中估计的变点选择。风险膨胀也将用来衡量偏差使用变量选择程序结合启发式搜索方法。还将开发一种新的使用自适应维度惩罚的变量选择程序。现代统计方法的主要目标之一是提供将输入变量与感兴趣的输出关联起来的统计模型。例如，这些模型被用来预测和解释农业的年作物产量、商业和经济的利率、医学的癌症发病率和社会学的犯罪率。建立这种模型的一个关键组成部分是选择包含强预测和解释信息的输入变量。由于最近包含大量潜在输入变量的大型数据库的激增，这个组件尤其重要。所提出的研究将提供强大的新方法来选择这些输入变量，用于各种各样的模型构建情况。

The proposed research will develop and evaluate variable selection procedures for multiple regression and related problems. Both the Bayesian and the frequentist points of view will be considered. From the Bayesian perspective, the focus will be on stochastic search variable selection methods which use a hierarchical mixture model to guide variable selection. These methods will include a procedure for fast Bayes variable selection to handle hundreds of variables at once, a procedure for variable selection in generalized linear models, a procedure for variable selection across exchangeable regressions, and a procedure for performing simultaneous variable selection and outlier removal. From the frequentist perspective, the focus will be on a new criterion for evaluating selection criteria called risk inflation. This research will entail the application and generalization of risk inflation to a broad set of model selection problems which includes generalized linear models, order selection procedures for time series models, and change-point selection for estimation in change-point problems. Risk inflation will also be used to gauge the bias of using variable selection procedures in conjunction with heuristic search methods. A new class of variable selection procedures which use adaptive dimensionality penalties will also be developed. One of the main goals of modern statistical methods is to provide statistical models which relate input variables to outputs of interest. For example, such models are used to predict and explain annual crop production in agriculture, interest rates in business and economics, cancer incidence in medicine, and crime rates in sociology. A key component in building such models is the selection of input variables which contain strong predictive and explanatory information. This component is especially important because of the recent proliferation of large databases containing vast numbers of potential input variables. The propos ed research will provide powerful new methods for selecting such input variables for a wide variety of model building situations.