## Statistics Papers

#### Title

Inequalities and Positive-Definite Functions Arising From a Problem in Multidimensional Scaling

#### Document Type

Journal Article

#### Date of this Version

1994

#### Publication Source

The Annals of Statistics

#### Volume

22

#### Issue

1

#### Start Page

406

#### Last Page

438

#### DOI

10.1214/aos/1176325376

#### Abstract

We solve the following variational problem: Find the maximum of ** E ∥ X−Y ∥** subject to

**, where**

*E*∥*X*∥^{2 }≤ 1**and**

*X**are i.i.d. random*

**Y****-vectors, and ∥⋅∥ is the usual Euclidean norm on**

*n***R**. This problem arose from an investigation into multidimensional scaling, a data analytic method for visualizing proximity data. We show that the optimal

^{n}**is unique and is (1) uniform on the surface of the unit sphere, for dimensions**

*X***, (2) circularly symmetric with a scaled version of the radial density**

*n*≥ 3**, for**

*ρ*/(1−*ρ*^{2})^{1/2}, 0 ≤*ρ*≤1**, and (3) uniform on an interval centered at the origin, for**

*n*=2**(Plackett's theorem). By proving spherical symmetry of the solution, a reduction to a radial problem is achieved. The solution is then found using the Wiener-Hopf technique for (real)**

*n*=1**. The results are reminiscent of classical potential theory, but they cannot be reduced to it. Along the way, we obtain results of independent interest: for any i.i.d. random**

*n*< 3**-vectors**

*n***and**

*X*

**Y**,**E****∥**Further, the kernel

*∥**X*−*Y**≤*∥*E**∥.**X*+*Y***and**

*K*_{p},*β*(*x*,*y*) = ∥*x*+*y*∥*− ∥*^{β}_{p}*x*−*y*∥*,*^{β}_{p}*x*,*y*∈R^{n}**∥**, is positive-definite, that is, it is the covariance of a random field,

*x*∥*p*=(∑|*x*_{i}|^{p})^{1/p}**for some real-valued random process**

*K*_{p,β}(*x*,*y*) =*E*[*Z*(*x*)*Z*(*y*) ]**, for**

*Z*(*x*)**1 ≤**and

*p*≤ 2**0 <**(but not for

*β*≤*p*≤ 2**or**

*β*>*p***in general). Although this is an easy consequence of known results, it appears to be new in a strict sense. In the radial problem, the average distance**

*p>2***between two spheres of radii**

*D*(*r*_{1},*r*_{2})**and**

*r*_{1}**is used as a kernel. We derive properties of**

*r*_{2}**, including nonnegative definiteness on signed measures of zero integral.**

*D*(*r*_{1},*r*_{2})#### Keywords

multidimensional scaling, maximal expected distance, potential theory, inequalities, positive-definite functions, Wiener-Hopf technique

#### Recommended Citation

Buja, A.,
Logan, B. F.,
Reeds, J. A.,
&
Shepp, L. A.
(1994).
Inequalities and Positive-Definite Functions Arising From a Problem in Multidimensional Scaling.
*The Annals of Statistics,*
*22*
(1),
406-438.
http://dx.doi.org/10.1214/aos/1176325376

**Date Posted:** 27 November 2017

This document has been peer reviewed.