A general framework for the asymptotic validity of two-stage procedures for selection and multiple comparisons with consistent variance estimators

We consider two-stage procedures for selection and multiple comparisons, where the variance parameter is estimated consistently. We examine conditions under which the procedures are asymptotically valid in a general framework. Our proofs of asymptotic validity require that the estimators at the end of the second stage are asymptotically normal, so we require a random-time-change central limit theorem. We explain how the assumptions hold for comparing means in transient simulations, steady-state simulations and quantile estimation, but the assumptions are also valid for many other problems arising in simulation studies.