##
**Fast large-sample goodness-of-fit tests for copulas.**
*(English)*
Zbl 1214.62049

Summary: Goodness-of-fit tests are a fundamental element in the copula-based modeling of multivariate continuous distributions. Among the different procedures proposed in the literature, recent large scale simulations suggest that one of the most powerful tests is based on the empirical process comparing the empirical copula with a parametric estimate of the copula derived under the null hypothesis. As for most of the currently available goodness-of-fit procedures for copula models, the null distribution of the statistic for the latter test is obtained by means of a parametric bootstrap. The main inconvenience of this approach is its high computational cost, which, as the sample size increases, can be regarded as an obstacle to its application.

In this work, fast large-sample tests for assessing goodness of fit are obtained by means of multiplier central limit theorems. The resulting procedures are shown to be asymptotically valid when based on popular method-of-moment estimators. Large scale Monte Carlo experiments, involving six frequently used parametric copula families and three different estimators of the copula parameter, confirm that the proposed procedures provide a valid, much faster alternative to the corresponding parametric bootstrap-based tests. An application of the derived tests to the modeling of a well-known insurance data set is presented. The use of the multiplier approach instead of the parametric bootstrap can reduce the computing time from about a day to minutes.

In this work, fast large-sample tests for assessing goodness of fit are obtained by means of multiplier central limit theorems. The resulting procedures are shown to be asymptotically valid when based on popular method-of-moment estimators. Large scale Monte Carlo experiments, involving six frequently used parametric copula families and three different estimators of the copula parameter, confirm that the proposed procedures provide a valid, much faster alternative to the corresponding parametric bootstrap-based tests. An application of the derived tests to the modeling of a well-known insurance data set is presented. The use of the multiplier approach instead of the parametric bootstrap can reduce the computing time from about a day to minutes.

### MSC:

62G10 | Nonparametric hypothesis testing |

62H05 | Characterization and structure theory for multivariate probability distributions; copulas |

60F05 | Central limit and other weak theorems |

62G30 | Order statistics; empirical distribution functions |

62H15 | Hypothesis testing in multivariate analysis |

62E15 | Exact distribution theory in statistics |

65C60 | Computational problems in statistics (MSC2010) |

62F40 | Bootstrap, jackknife and other resampling methods |

62P05 | Applications of statistics to actuarial sciences and financial mathematics |

65C05 | Monte Carlo methods |