3. Multivariate Extremes: A Conditional Quantile Approach

There is no natural ordering of a multidimensional space and the extension of the definition of univariate quantiles to multivariate distributions is, therefore, not straightforward. We use a definition based on the ordering of a multivariate sample according to an increasing family of curves, we have called isobars. For a given level u, a u-level isobar is defined as a level curve of the conditional distribution function of the radius given the angle (that is, as a conditional quantile), the sample points being defined by their polar coordinates. In this way, the maximum value of the sample is defined as the point which belongs to the upper level isobar and it is really a sample point. Moreover, the so-defined order statistics may be characterized by a unidimensional approach. First we recall some results concerning the weak stability of these extreme values. Furthermore other applications are here proposed as the definition of the corresponding record values for a multivariate distribution and the stability properties of this kind of record values.