  In this article, we introduce the concept of model confidence bounds (MCB)
for variable selection in the context of nested models. Similarly to the
endpoints in the familiar confidence interval for parameter estimation, the MCB
identifies two nested models (upper and lower confidence bound models)
containing the true model at a given level of confidence. Instead of trusting a
single selected model obtained from a given model selection method, the MCB
proposes a group of nested models as candidates and the MCB's width and
composition enable the practitioner to assess the overall model selection
uncertainty. A new graphical tool --- the model uncertainty curve (MUC) --- is
introduced to visualize the variability of model selection and to compare
different model selection procedures. The MCB methodology is implemented by a
fast bootstrap algorithm that is shown to yield the correct asymptotic coverage
under rather general conditions. Our Monte Carlo simulations and real data
examples confirm the validity and illustrate the advantages of the proposed
method.
