Multidimensional outlier-proneness of dependent data and the extremal index

Let (Θn,Rn)(Θn,Rn) be the polar representation of Xn,n≥1, with values in RdRd. We quantify the outlier-proneness of the sequence X={Xn}n≥1 by ordering its variables according to the values of FΘn(Rn),n≥1FΘn(Rn),n≥1, and by defining two sequences of outlier-proneness coefficients. Such coefficients take into account the clustering of probability level surfaces Fθ−1(FΘn(Rn)), containing or contained in normalized probability level surfaces for the underlying distribution of X, and let us view the extremal index as an indicator of the outlier-proneness of a multidimensional sequence.We illustrate the results with bivariate Gaussian processes with different outlier-proneness.