In testing that a particular distribution belongs to a parameterized family , one often compares a non-parametric estimate of , with a member of . In most cases, the limiting distribution of goodness-of-fit statistics based on depends on the unknown distribution . It is shown here that if the sequence () of estimators is regular in some sense, then the parametric bootstrap approach is valid, i.e., if and are analogs of and calculated from a bootstrap sample from , then the empirical processes and converge jointly to independent and identically distributed limits. These results are used to establish the validity of the parametric bootstrap method when testing the goodness-of-fit of parametric families of multivariate distributions and copulas. Two types of tests are considered: those based on a distance between an empirical multivariate distribution function or copula and its parametric estimation under the null hypothesis, and those based on a distance between empirical and parametric estimations of univariate pseudo-observations obtained via a probability integral transformation. For situations in which the multivariate distribution function or its probability integral transformation cannot be obtained in closed form, a two-level parametric bootstrap is developed and its validity is established. As an illustration, the one- and two-level parametric bootstrap methodology is detailed in the special case of a goodness-of-fit test statistic for copula models based on a Cram´er–von Mises type functional of the empirical copula process.
Published July 2005 , 42 pages