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Asymptotically distribution-free goodness-of-fit testing for tail copulas. (English) Zbl 1312.62072

Summary: Let \((X_{1},Y_{1}),\dots,(X_{n},Y_{n})\) be an i.i.d. sample from a bivariate distribution function that lies in the max-domain of attraction of an extreme value distribution. The asymptotic joint distribution of the standardized component-wise maxima \(\bigvee_{i=1}^{n}X_{i}\) and \(\bigvee_{i=1}^{n}Y_{i}\) is then characterized by the marginal extreme value indices and the tail copula \(R\). We propose a procedure for constructing asymptotically distribution-free goodness-of-fit tests for the tail copula \(R\). The procedure is based on a transformation of a suitable empirical process derived from a semi-parametric estimator of \(R\). The transformed empirical process converges weakly to a standard Wiener process, paving the way for a multitude of asymptotically distribution-free goodness-of-fit tests. We also extend our results to the \(m\)-variate (\(m>2\)) case. In a simulation study we show that the limit theorems provide good approximations for finite samples and that tests based on the transformed empirical process have high power.

MSC:

62H15 Hypothesis testing in multivariate analysis
62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
62G32 Statistics of extreme values; tail inference
62F03 Parametric hypothesis testing
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