Valid Confidence Intervals In Regression After Variable Selection

We consider a linear regression model with regression parameters (θ1,...,θp) and error variance parameter σ2. Our aim is to find a confidence interval with minimum coverage probability 1 − α for a parameter of interest θ1 in the presence of nuisance parameters (θ2,...,θp,σ2). We consider two confidence intervals, the first of which is the standard confidence interval for θ1 with coverage probability 1 − α. The second confidence interval for θ1 is obtained after a variable selection procedure has been applied to θp. This interval is chosen to be as short as possible subject to the constraint that it has minimum coverage probability 1 − α. The confidence intervals are compared using a risk function that is defined as a scaled version of the expected length of the confidence interval. We show that, subject to certain conditions including that [(dimension of response vector) − p] is small, the second confidence interval is preferable to the first when we anticipate (without being certain) that |θp|/σ is small. This comparison of confidence intervals is shown to be mathematically equivalent to a corresponding comparison of prediction intervals.