The Chebyshev center: A multidimensional estimate of location

Abstract

In this paper the center of the smallest k-dimensional sphere enclosing a data set, hereinafter called the Chebyshev center, is introduced as a multidimensional measure of location. The distribution of this estimator, which is a multidimensional generalization of the univariate midrange, is derived in the general case and its properties investigated for a host of distributions. The Chebyshev center is shown to be a maximum likelihood estimator for the center of a uniform distribution over a k-sphere and both unbiased and consistent for the multivariate spherical normal distribution and any spherical finite range distribution.