A unified approach to proving parametric bootstrap consistency for some goodness-of-fit tests

Because model misspecification can lead to inconsistent and inefficient estimators and invalid tests of hypotheses, testing for misspecification is critically important. We focus here on several general purpose goodness-of-fit tests which can be applied to assess the adequacy of a wide variety of parametric models without specifying an alternative model. Parametric bootstrap is the method of choice for computing the p-values of these tests however the proof of its consistency has never been rigourously shown in this setting. Using properties of locally asymptotically normal parametric models, we prove that under quite general conditions, the parametric bootstrap provides a consistent estimate of the null distribution of the statistics under investigation.