## The Annals of Statistics

- Ann. Statist.
- Volume 22, Number 1 (1994), 406-438.

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

Andreas Buja, B. F. Logan, J. A. Reeds, and L. A. Shepp

#### Abstract

We solve the following variational problem: Find the maximum of $E\|X - Y\|$ subject to $E\|X\|^2 \leq 1$, where $X$ and $Y$ are i.i.d. random $n$-vectors, and $\|\cdot\|$ is the usual Euclidean norm on $\mathbb{R}^n$. This problem arose from an investigation into multidimensional scaling, a data analytic method for visualizing proximity data. We show that the optimal $X$ is unique and is (1) uniform on the surface of the unit sphere, for dimensions $n \geq 3$, (2) circularly symmetric with a scaled version of the radial density $\rho/(1 - \rho^2)^{1/2}, 0 \leq \rho \leq 1$, for $n = 2$, and (3) uniform on an interval centered at the origin, for $n = 1$ (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) $n < 3$. 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 $n$-vectors $X$ and $Y, E\|X - Y\| \leq E\|X + Y\|$. Further, the kernel $K_{p,\beta}(x, y) = \|x + y\|^\beta_p - \|x - y\|^\beta_p, x, y \in \mathbb{R}^n$ and $\|x\|p = (\sum|x_i|^p)^{1/p}$, is positive-definite, that is, it is the covariance of a random field, $K_{p,\beta}(x, y) = E\lbrack Z(x)Z(y)\rbrack$ for some real-valued random process $Z(x)$, for $1 \leq p \leq 2$ and $0 < \beta \leq p \leq 2$ (but not for $\beta > p$ or $p > 2$ 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 $D(r_1, r_2)$ between two spheres of radii $r_1$ and $r_2$ is used as a kernel. We derive properties of $D(r_1, r_2)$, including nonnegative definiteness on signed measures of zero integral.

#### Article information

**Source**

Ann. Statist., Volume 22, Number 1 (1994), 406-438.

**Dates**

First available in Project Euclid: 11 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176325376

**Digital Object Identifier**

doi:10.1214/aos/1176325376

**Mathematical Reviews number (MathSciNet)**

MR1272091

**Zentralblatt MATH identifier**

0834.62060

**JSTOR**

links.jstor.org

**Subjects**

Primary: 26D10: Inequalities involving derivatives and differential and integral operators

Secondary: 62H99: None of the above, but in this section 42A82: Positive definite functions 45E10: Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) [See also 47B35]

**Keywords**

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

#### Citation

Buja, Andreas; Logan, B. F.; Reeds, J. A.; Shepp, L. A. Inequalities and Positive-Definite Functions Arising from a Problem in Multidimensional Scaling. Ann. Statist. 22 (1994), no. 1, 406--438. doi:10.1214/aos/1176325376. https://projecteuclid.org/euclid.aos/1176325376