Local robust estimation of the Pickands dependence function
Let \(\left( Y^{(1)},Y^{(2)} \right) \) be a pair of random variables which is recorded along with a random covariate \( X \in \mathbb{R}^{p} \). Denote by \( C_{x}, F_{1}\left(\cdot | x \right) \) and \( F_{2}\left(\cdot | x \right) \) the conditional copula function and the continuous distribution functions of \( Y^{1} \) and \( Y^{2} \) given \( X=x \). The authors consider the model which is written as \[ \mathbb{P}\left(F_{1} \left( Y^{1} | x \right) \leq y_{1}, \quad F_{2} \left( Y^{2} | x \right) \leq y_{2} \right) = C_{x}\left( y_{1}, y_{2} \right), \] where the conditional extreme value copula \( C_{x} \) admits a representation of the form \[ C_{x}\left( y_{1},y_{2} \right) = \exp \left( \log\left( y_{1}y_{2} \right) A\left( \frac{\log\left( y_{2} \right)}{\log\left( y_{1}y_{2} \right)} \Big| x \right) \right), \] and \( A\left(\cdot|\cdot \right) : [0,1] \times \mathbb{R}^{p} \to [1/2,1]\) is the conditional Pickands dependence function. The authors' abstract: We consider the robust estimation of the Pickands dependence function in the random covariate framework. Our estimator is based on local estimation with the minimum density power divergence criterion. We provide the main asymptotic properties, in particular the convergence of the stochastic process, correctly normalized, towards a tight centered Gaussian process. The finite sample performance of our estimator is evaluated with a simulation study involving both uncontaminated and contaminated samples. The method is illustrated on a dataset of air pollution measurements.
conditional Pickands dependence function
scientific article; zbMATH DE number 6968600
scientific article