标题： 回归后初步F检验的置信区间覆盖率概率 显示英文标题

标题： The coverage probabililty of confidence intervals in regression after a preliminary F test

摘要： 考虑一个线性回归模型，回归参数为beta=(beta_1,..., beta_p)，独立正态误差。 假设感兴趣的参数是theta = a^T beta，其中a是给定的。 定义s维参数向量tau = C^T beta - t，其中C和t是给定的。 假设我们进行一个初步的F检验，检验原假设H_0: tau = 0与备择假设H_1: tau不等于0。 通常的统计实践是然后构造一个名义覆盖概率为1-alpha的theta的置信区间，使用相同的数据，基于所选模型已经被事先给出（作为真实模型）的假设。 我们将这称为theta的naive 1-alpha置信区间。 这个假设是错误的，可能导致这个置信区间的最小覆盖概率远低于1-alpha，使其完全不适用。 我们的目标是计算这个最小覆盖概率。 很容易找到这个置信区间的覆盖概率的表达式，这是一个s+1维的多重积分。 然而，我们推导出一个新的优雅且计算方便的公式来表示这个覆盖概率。 对于s=2，这个公式是一个三重积分和一个双重积分的和，而对于所有s>2，这个公式是一个四重积分和一个双重积分的和。 这使得无论s有多大，计算naive置信区间的最小覆盖概率变得容易。 这个公式的一个非常重要的实际应用是协方差分析。 在此背景下，可以定义tau使得H_0表达了"平行性"的假设。 应用统计学家通常建议进行这个假设的初步F检验。 我们通过一个实际的协方差数据分析集和一个"平行性"的初步F检验来说明我们公式的应用。 我们表明，naive 0.95置信区间具有最小覆盖概率0.0846，这表明它是完全不适用的。 显示英文摘要

摘要： Consider a linear regression model with regression parameter beta=(beta_1,..., beta_p) and independent normal errors. Suppose the parameter of interest is theta = a^T beta, where a is specified. Define the s-dimensional parameter vector tau = C^T beta - t, where C and t are specified. Suppose that we carry out a preliminary F test of the null hypothesis H_0: tau = 0 against the alternative hypothesis H_1: tau not equal to 0. It is common statistical practice to then construct a confidence interval for theta with nominal coverage 1-alpha, using the same data, based on the assumption that the selected model had been given to us a priori(as the true model). We call this the naive 1-alpha confidence interval for theta. This assumption is false and it may lead to this confidence interval having minimum coverage probability far below 1-alpha, making it completely inadequate. Our aim is to compute this minimum coverage probability. It is straightforward to find an expression for the coverage probability of this confidence interval that is a multiple integral of dimension s+1. However, we derive a new elegant and computationally-convenient formula for this coverage probability. For s=2 this formula is a sum of a triple and a double integral and for all s>2 this formula is a sum of a quadruple and a double integral. This makes it easy to compute the minimum coverage probability of the naive confidence interval, irrespective of how large s is. A very important practical application of this formula is to the analysis of covariance. In this context, tau can be defined so that H_0 expresses the hypothesis of "parallelism". Applied statisticians commonly recommend carrying out a preliminary F test of this hypothesis. We illustrate the application of our formula with a real-life analysis of covariance data set and a preliminary F test for "parallelism". We show that the naive 0.95 confidence interval has minimum coverage probability 0.0846, showing that it is completely inadequate.