Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T18:20:16.955Z Has data issue: false hasContentIssue false
  • English
  • Français

A classification theorem for planar distributions based on the shape statistics of independent tetrads

Published online by Cambridge University Press:  24 October 2008

Christopher G. Small
Christopher G. Small
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ont. N2L 3Gl, Canada
Get access Rights & Permissions [Opens in a new window]

Extract

The uniformity of a certain shape density of a random ΠD tetrad relative to Lebesgue measure along part of the set corresponding to alignment is shown to be sufficient under certain technical conditions for the generating distribution to be the uniform distribution on a compact convex set. A counter-example is provided to show that this result fails for random IID triangles. The main theorem is used to characterize circular, elliptical and triangular uniform generators from the shape distribution of random IID tetrads.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 96 , Issue 3 , November 1984 , pp. 543 - 547
Copyright
Copyright © Cambridge Philosophical Society 1984

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Blaschke, W.. Vorlesungen uber Differentialgeometrie, vol. 2, Affine Differentialgeometrie (Springer, 1923).Google Scholar
[2]Groemer, H.. On some mean values associated with a randomly selected simplex in a convex set. Pacific J. Math. 45 (1973), 525–533.CrossRefGoogle Scholar
[3]Kendall, D. G.. Shape-manifolds, procrustean metrics and complex projective spaces. Bull. London Math. Soc. 16 (1984), 81–121.CrossRefGoogle Scholar
[4]Mallows, C. L. and Clark, J. M.. Linear intercept distributions do not characterize plane sets. J. Appl. Probab. 7 (1970), 240–244.CrossRefGoogle Scholar
[5]Santalo, L. A.. Integral Geometry and Geometric Probability (Addison-Wesley, 1976).Google Scholar
[6]Small, C. G.. Random uniform triangles and the alignment problem. Math. Proc. Cambridge Philos. Soc. 91 (1982), 315–322.CrossRefGoogle Scholar
[7]Small, C. G.. Characterization of distributions from maximal invariant statistics. Z. Wahrsch. Verw. Gebiete 64 (1983), 517–527.CrossRefGoogle Scholar