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ON THE DISPERSION OF MULTIVARIATE MEDIAN

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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.

**Source**
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