On scale curves for nonparametric description of dispersion

A two-dimensional visual device for description of the dispersion of a multivariate data set in any dimension is the scale curve, a depth-based method introduced by Liu (1990) and treated in detail by Liu, Parelius and Singh (1999). It offers an appealing alternative to covariance matrix methods and even provides an attractive new tool in the univariate case. To support applications such as uniform confidence bounds for the population scale curve, we develop some asymptotics. Uniform strong convergence of the sample scale curve is established under broad conditions and uniform weak convergence under somewhat restricted conditions. Open issues for further investigation are discussed.