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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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