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Goodness-of-fit testing for copulas: a distribution-free approach. (English) Zbl 1471.62357

For a \(d\) dimensional random variable, it is known that its distribution function \(F\) can be written as a function of its \(d\) marginal distributions. This function is called the copula associated with \(F\) and, in some practical applications, it is of interest in testing a null hypothesis that the copula is from a specific parametric family. Under an assumption that the marginal distributions of \(F\) are from a parametric family, this paper proposes to test this goodness-of-fit hypothesis using a functional of empirical process on the unit hypercube defined based on an empirical version of the transformation on the difference between a semiparametric estimate and a parametric estimate of the copula. The asymptotic distribution of this family of tests is shown to be distribution-free under the null hypothesis and asymptotic optimal under a contiguous alternative hypothesis. The proposed tests are evaluated through simulation studies and illustrated through a real data analysis.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
60F17 Functional limit theorems; invariance principles
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References:

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