标题： 变量选择后回归中置信区间的最小覆盖概率的上界 显示英文标题

标题： Upper bounds on the minimum coverage probability of confidence intervals in regression after variable selection

摘要： 我们考虑一个线性回归模型，感兴趣的参数是回归参数向量的指定线性组合。假设首先通过数据驱动的模型选择方法（例如，通过初步假设检验或最小化AIC）来选择模型。通常的统计实践是基于所选模型是先验给定的假设，构建感兴趣参数的置信区间。然而，这个假设是错误的，并可能导致置信区间具有较差的覆盖率性质。我们提供了一个易于计算的有限样本上界（通过重复数值评估双重积分计算得出），用于该置信区间的最低覆盖率概率。此上界适用于以下任意一种模型选择方法：最小AIC、最小BIC、最大调整R平方、最小Mallows' Cp和t检验。这一上界的重要意义在于它界定了设计矩阵和模型选择程序的一般类别，对于这些类别，该置信区间具有较差的覆盖率性质。此上界被证明是Kabaila和Leeb早期大样本上界的有限样本类比。 显示英文摘要

摘要： We consider a linear regression model, with the parameter of interest a specified linear combination of the regression parameter vector. We suppose that, as a first step, a data-based model selection (e.g. by preliminary hypothesis tests or minimizing AIC) is used to select a model. It is common statistical practice to then construct a confidence interval for the parameter of interest based on the assumption that the selected model had been given to us a priori. This assumption is false and it can lead to a confidence interval with poor coverage properties. We provide an easily-computed finite sample upper bound (calculated by repeated numerical evaluation of a double integral) to the minimum coverage probability of this confidence interval. This bound applies for model selection by any of the following methods: minimum AIC, minimum BIC, maximum adjusted R-squared, minimum Mallows' Cp and t-tests. The importance of this upper bound is that it delineates general categories of design matrices and model selection procedures for which this confidence interval has poor coverage properties. This upper bound is shown to be a finite sample analogue of an earlier large sample upper bound due to Kabaila and Leeb.