NDLI: Some Problems in Probabilistic Tomography

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Applegate, D. Shor, P. Scheinberg, S. Shepp, L. Reeds, J.
Copyright Year	1997
Abstract	Given probability distributions $F_1 , F_2 , \ldots , F_k$ on $\bR$ and distinct directions $\theta_1 \!, \ldots , \theta_k\!$, one may ask whetherthere is a probability measure $\mu$ on $\bR^2$ such that the marginal of $\mu$ in direction $\theta_j$ is $F_j$, $j=1, \ldots , k$. For example for $k=3$ we ask what the marginal of $\mu$ at $45^\circ$ can be if the x and y marginals are each say standard normal? In probabilistic language, if X and Y are each standard normal with an {\it arbitrary} joint distribution, what can the distribution of $X+Y$ or $X-Y$ be? This type of question is familiar to probabilists and is also familiar (except perhaps in that $\mu$ is positive) to tomographers, but is difficult to answer in special cases. The set of distributions for $Z = X-Y$ is a convex and compact set, C, which contains the single point mass $Z \equiv 0$ since $X \equiv Y$, standard normal, is possible. We show that Z can be 3-valued, $Z=0$, $\pm a$ for any a, each with positive probability, but Z {\it cannot} have any (genuine) two-point distribution. Using numerical linear programming we present convincing evidence that Z can be uniform on the interval $[- \ve, \ve]$ for $\ve$ small and give estimates for the largest such $\ve$. The set of all extreme points of C seems impossible to determine explicitly. We also consider the more basic question of finding the extreme measures on the unit square with uniform marginals on both coordinates, and show that not every such measure has a support which has only one point on each horizontal or vertical line, which seems surprising.}
Starting Page	199
Ending Page	209
Page Count	11
File Format	PDF
ISSN	0040585X
DOI	10.1137/S0040585X97975435
e-ISSN	10957219
Journal	Theory of Probability & Its Applications (TPRBAU)
Issue Number	2
Volume Number	41
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-25
Access Restriction	Subscribed
Subject Keyword	marginal distributions Radon extreme point
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Statistics, Probability and Uncertainty

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