ON THE DISPERSION OF MULTIVARIATE MEDIAN

The estimation of the asymptotic variance of sample median based on a random sample of univariate observations has been extensively studied in the literature. The appearance of a ''local object'' like the densi ty function of the observations in this asymptotic variance makes its estimation a difficult task, and there are several complex technical p roblems associated with it. This paper explores the problem of estimat ing the dispersion matrix of the multivariate L1 median. Though it is absolutely against common intuition, this problem turns out to be tech nically much simpler. We exhibit a simple estimate for the large sampl e dispersion matrix of the multivariate L1 median with excellent asymp totic properties. and to construct this estimate, we do not use any of the computationally intensive resampling techniques (e.g. the general ized jack-knife, the bootstrap. etc. that have been used and thoroughl y investigated by leading statisticians in their attempts to estimate the asymptotic variance of univariate median). However surprising may it sound, our analysis exposes that most of the technical complicacies associated with the estimation of the sampling variation in the media n are only characteristics of univariate data, and they disappear as s oon as we enter into the realm of multivariate analysis.