This article is available for purchase or by subscription. See below.
Abstract
|
It is known that spheres have negative type, but only subsets with at most one pair
of antipodal points have strict negative type. These are conditions on the (angular)
distances within any finite subset of points. We show that subsets with at most one
pair of antipodal points have strong negative type, a condition on every probability
distribution of points. This implies that the function of expected distances to points
determines uniquely the probability measure on such a set. It also implies that the
distance covariance test for stochastic independence, introduced by Székely, Rizzo
and Bakirov, is consistent against all alternatives in such sets. Similarly, it allows
tests of goodness of fit, equality of distributions, and hierarchical clustering with
angular distances. We prove this by showing an analogue of the Cramér–Wold
theorem.
|
PDF Access Denied
We have not been able to recognize your IP address
44.222.161.54
as that of a subscriber to this journal.
Online access to the content of recent issues is by
subscription, or purchase of single articles.
Please contact your institution's librarian suggesting a subscription, for example by using our
journal-recommendation form.
Or, visit our
subscription page
for instructions on purchasing a subscription.
You may also contact us at
contact@msp.org
or by using our
contact form.
Or, you may purchase this single article for
USD 40.00:
Keywords
Cramér–Wold, hemispheres, expected distances, distance
covariance, equality of distributions, goodness of fit,
hierarchical clustering.
|
Mathematical Subject Classification 2010
Primary: 44A12, 45Q05, 51K99, 51M10
Secondary: 62G20, 62H15, 62H20, 62H30
|
Milestones
Received: 16 May 2019
Revised: 4 April 2020
Accepted: 19 May 2020
Published: 4 September 2020
|
|