On spatial quantiles (Q2726264)

It is well known that in one dimension for any probability distribution \(P\) with a finite first moment the medians are exactly the values of \(m\) for which \(\int|x-m|P(dx)\) is minimized. This characterization allows the definition of median to be extended to more than one dimension -- spatial medians. Let \(P\) be a probability in \(R^{d}\) and let \(G_{P}\) be the map from \(R^{d}\) into \(R^{d}\) defined by NEWLINE\[NEWLINEG_{P}(s)=\int_{x\not= s}(s-x)|s-x|^{-1}P(dx),NEWLINE\]NEWLINE where \(|\cdot|\) is the usual Euclidean norm. It is shown that \(G_{P}\) is a homeomorphism of \(R^{d}\) onto the open unit ball with center \(0\) if \(P\) is not concentrated in a line and is nonatomic. If \(P\) has atoms, but still is not concentrated in a line, then the range of \(G_{P}\) has a more complicated structure. In the case of spherically symmetric \(P\) a representation of the map \(G_{P}\) is given involving just one distribution of a positive random variable. Asymptotic properties of empirical spatial quantiles are also given (under some differentiability of \(G_{P}\)).NEWLINENEWLINEFor the entire collection see [Zbl 0956.00022].