Title: The Distance Standard Deviation,
Abstract: The distance standard deviation, which arises in distance correlation
analysis of multivariate data, is studied as a measure of spread. New
representations for the distance standard deviation are obtained in terms of
Gini's mean difference and in terms of the moments of spacings of order
statistics. Inequalities for the distance variance are derived, proving that
the distance standard deviation is bounded above by the classical standard
deviation and by Gini's mean difference. Further, it is shown that the distance
standard deviation satisfies the axiomatic properties of a measure of spread.
Explicit closed-form expressions for the distance variance are obtained for a
broad class of parametric distributions. The asymptotic distribution of the
sample distance variance is derived.