Groupe d’études et de recherche en analyse des décisions

# Validity of the Parametric Bootstrap for Goodness-of-Fit Testing in Semiparametric Models

## Christian Genest et Bruno Rémillard

In testing that a particular distribution $P$ belongs to a parameterized family $\cal{P}$, one often compares a non-parametric estimate $P_n$ of $P$, with a member $P_\theta_n$ of $\cal{P}$. In most cases, the limiting distribution of goodness-of-fit statistics based on $n^{1/2}(P_n-P_\theta_n)$ depends on the unknown distribution $P$. It is shown here that if the sequence ($\theta_n$) of estimators is regular in some sense, then the parametric bootstrap approach is valid, i.e., if $P^*_n$ and $\theta^*_n$ are analogs of $P_n$ and $\theta_n$ calculated from a bootstrap sample from $P_\theta_n$, then the empirical processes $n^{1/2}(P_n-P_\theta_n)$ and $n^{1/2}(P^*_n-P_{\theta^*_n})$ 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.

, 42 pages