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|a (JST)2242461
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|a eng
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|2 MSC
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|2 MSC
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|a Buja, Andreas
|e verfasserin
|4 aut
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|a Inequalities and Positive-Definite Functions Arising from a Problem in Multidimensional Scaling
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|c 1994
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|a Text
|b txt
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|a Computermedien
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|a Online-Ressource
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|a We solve the following variational problem: Find the maximum of E|X - Y| subject to E|X|2≤ 1, where X and Y are i.i.d. random n-vectors, and |·| is the usual Euclidean norm on Rn. 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 ≥ 3, (2) circularly symmetric with a scaled version of the radial density ρ/(1 - ρ2)1/2, 0 ≤ ρ ≤ 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| ≤ E|X + Y|. Further, the kernel Kp,β(x, y) = |x + y|β p- |x - y|β p, x, y ∈ Rnand |x|p = (∑|xi|p)1/p, is positive-definite, that is, it is the covariance of a random field, Kp,β(x, y) = E[ Z(x)Z(y)] for some real-valued random process Z(x), for 1 ≤ p ≤ 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(r1, r2) between two spheres of radii r1and r2is used as a kernel. We derive properties of D(r1, r2), including nonnegative definiteness on signed measures of zero integral.
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|a Copyright 1994 Institute of Mathematical Statistics
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|a Multidimensional scaling
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|a maximal expected distance
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|a potential theory
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|a inequalities
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|a positive-definite functions
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|a Wiener-Hopf technique
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|a Physical sciences
|x Physics
|x Mechanics
|x Mass
|x Point masses
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| 650 |
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|a Mathematics
|x Mathematical expressions
|x Mathematical functions
|x Mathematical transformations
|x Integral transformations
|x Laplace transformation
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|a Information science
|x Information management
|x Data management
|x Data visualization
|x Multidimensional scaling
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| 650 |
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|a Mathematics
|x Pure mathematics
|x Calculus
|x Differential calculus
|x Differential equations
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| 650 |
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|a Mathematics
|x Mathematical expressions
|x Mathematical functions
|x Stress functions
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| 650 |
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|a Mathematics
|x Applied mathematics
|x Analytics
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| 650 |
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|a Mathematics
|x Pure mathematics
|x Probability theory
|x Random variables
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| 650 |
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|a Behavioral sciences
|x Leisure studies
|x Recreation
|x Sports
|x Equestrianism
|x Horse tack
|x Horseshoes
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| 650 |
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|a Mathematics
|x Mathematical objects
|x Mathematical series
|x Series convergence
|x Absolute convergence
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| 650 |
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|a Mathematics
|x Applied mathematics
|x Statistics
|x Applied statistics
|x Descriptive statistics
|x Statistical distributions
|x Distribution functions
|x Probability distributions
|x Mathematical moments
|x Multivariate Analysis
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|a research-article
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|a Logan, B. F.
|e verfasserin
|4 aut
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|a Reeds, J. A.
|e verfasserin
|4 aut
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|a Shepp, L. A.
|e verfasserin
|4 aut
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|i Enthalten in
|t The Annals of Statistics
|d Institute of Mathematical Statistics
|g 22(1994), 1, Seite 406-438
|w (DE-627)270129162
|w (DE-600)1476670-X
|x 00905364
|7 nnns
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|g volume:22
|g year:1994
|g number:1
|g pages:406-438
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|u https://www.jstor.org/stable/2242461
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|d 22
|j 1994
|e 1
|h 406-438
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