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Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models. (English) Zbl 1206.62044
Summary: 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 \(A_{\theta _n}\) corresponding to an estimate \(\theta _n\) of \(\theta \). In many cases, the asymptotic distribution of goodness-of-fit statistics derived from the process \(n^{1/2}(A_n - A_{\theta_n})\) depends on the unknown distribution \(P\). It is shown here that if the sequences \(A_n\) and \(\theta _n\) 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 \(\theta _n^{*}\) are analogs of \(A_n\) and \(\theta _n\) computed from a sample from \(P_{\theta _n}\), the empirical processes \(n^{1/2}(A_n - A_{\theta _n})\) and \(n^{1/2}(A_n^{*} - A_{\theta _n^{*}})\) 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.

MSC:
62F40 Bootstrap, jackknife and other resampling methods
62F05 Asymptotic properties of parametric tests
62G10 Nonparametric hypothesis testing
62H15 Hypothesis testing in multivariate analysis
62G05 Nonparametric estimation
65C05 Monte Carlo methods
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