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.


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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