Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models

Genest, Christian ; Rémillard, Bruno

Ann. Inst. H. Poincaré Probab. Statist., Tome 44 (2008) no. 2, p. 1096-1127 / Harvested from Project Euclid

Résumé

In testing that a given distribution P belongs to a parameterized family $\mathcal{P}$ , one is often led to compare a nonparametric estimate A_n of some functional A of P with an element A_θn corresponding to an estimate θ_n of θ. In many cases, the asymptotic distribution of goodness-of-fit statistics derived from the process n^1/2(A_n−A_θn) depends on the unknown distribution P. It is shown here that if the sequences A_n and θ_n of estimators are regular in some sense, a parametric bootstrap approach yields valid approximations for the P-values of the tests. In other words if A_n^* and θ_n^* are analogs of A_n and θ_n computed from a sample from P_θn, the empirical processes n^1/2(A_n−A_θn) and n^1/2(A_n^*−A_θn^*) then converge jointly in distribution to independent copies of the same limit. This result is used to establish the validity of the parametric bootstrap method when testing the goodness-of-fit of families of multivariate distributions and copulas. Two types of tests are considered: certain procedures compare the empirical version of a distribution function or copula and its parametric estimation under the null hypothesis; others measure the distance between a parametric and a nonparametric estimation of the distribution associated with the classical probability integral transform. The validity of a two-level bootstrap is also proved in cases where the parametric estimate cannot be computed easily. The methodology is illustrated using a new goodness-of-fit test statistic for copulas based on a Cramér–von Mises functional of the empirical copula process.

Détail

Publié le : 2008-12-15
Classification:  Copula,  Goodness-of-fit test,  Monte Carlo simulation,  Parametric bootstrap,  P-values,  Semiparametric estimation,  62F05,  62F40,  62H15