This paper considers the construction of model selection procedures based on choosing the model with the largest maximised log-likelihood minus a penalty, when key parameters are restricted to be in a closed interval. The approach adopted is based on King et al.' s (1995) representative models method with the use of the parametric bootstrap to handle nuisance parameters. The method is illustrated by application to two model selection problems in the context of Box-Cox transformations and the linear regression model. Simulation results for both problems indicate that the new procedure clearly dominates existing procedures in terms of having higher probabilities of correctly selecting the true model.