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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.

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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