ON THE COVERAGE PROBABILITY OF CONFIDENCE INTERVALS IN REGRESSION AFTER VARIABLE SELECTION

摘要

This paper considers a linear regression model with regression parameter vector β. The parameter of interest is θ = a~Tβ where a is specified. When, as a first step, a data-based variable selection (e.g. minimum Akaike information criterion) is used to select a model, it is common statistical practice to then carry out inference about θ, using the same data, based on the (false) assumption that the selected model had been provided a priori. The paper considers a confidence interval for θ with nominal coverage 1- α constructed on this (false) assumption, and calls this the naive 1 - α confidence interval. The minimum coverage probability of this confidence interval can be calculated for simple variable selection procedures involving only a single variable. However, the kinds of variable selection procedures used in practice are typically much more complicated. For the real-life data presented in this paper, there are 20 variables each of which is to be either included or not, leading to 2~(20) different models. The coverage probability at any given value of the parameters provides an upper bound on the minimum coverage probability of the naive confidence interval. This paper derives a new Monte Carlo simulation estimator of the coverage probability, which uses conditioning for variance reduction. For these real-life data, the gain in efficiency of this Monte Carlo simulation due to conditioning ranged from 2 to 6. The paper also presents a simple one-dimensional search strategy for parameter values at which the coverage probability is relatively small. For these real-life data, this search leads to parameter values for which the coverage probability of the naive 0.95 confidence interval is 0.79 for variable selection using the Akaike information criterion and 0.70 for variable selection using Bayes information criterion, showing that these confidence intervals are completely inadequate.