ON THE DISPERSION OF MULTIVARIATE MEDIAN

ARUP BOSE AND PROBAL CHAUDHURI

Division of Theoretical Statistics and Mathematics, Indian Statistical Institute, 203 B. T. Road, Calcutta 700035, India

(Received November 28, 1991; revised August 18, 1992)

Abstract. 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 density function of the observations in this asymptotic variance makes its estimation a difficult task, and there are several complex technical problems associated with it. This paper explores the problem of estimating the dispersion matrix of the multivariate L_{1} median. Though it is absolutely against common intuition, this problem turns out to be technically much simpler. We exhibit a simple estimate for the large sample dispersion matrix of the multivariate L_{1} median with excellent asymptotic properties, and to construct this estimate, we do not use any of the computationally intensive resampling techniques (e.g. the generalized jackknife, the bootstrap, etc. that have been used and thoroughly 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 median are only characteristics of univariate data, and they disappear as soon as we enter into the realm of multivariate analysis.

Key words and phrases: Asymptotic dispersion matrix, consistent estimate, generalized variance, L_{1} median, multivariate Hodges-Lehmann estimate, n^{1/2}- consistent estimation, rate of convergence.