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

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