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The Mardia–Dryden shape distribution for triangles: a stochastic calculus approach

Published online by Cambridge University Press:  14 July 2016

David G. Kendall
David G. Kendall*
Affiliation:
University of Cambridge
*
∗ Postal address: Statistical Laboratory, 16 Mill Lane, Cambridge CB2 1SB, UK.
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Abstract

A remarkable distribution found by Mardia and Dryden for the shape of a random triangle in ℝ2 whose vertices are displaced from their initial positions by independent identical symmetrical gaussian perturbations is re-derived via stochastic calculus.

Keywords

BESSEL BRIDGEBESSEL PROCESSLEGENDRE POLYNOMIALRANDOM TRIANGLESIZE-AND-SHAPESPHERICAL BROWNIAN MOTIONWEBER EXPONENTIAL INTEGRAL

Information

Type
Short Communications
Information
Journal of Applied Probability , Volume 28 , Issue 1 , March 1991, pp. 225 - 230
Copyright
Copyright © Applied Probability Trust 1991 

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References

[1] Kendall, D. G. (1977) The diffusion of shape. Adv. Appl. Prob. 9, 428–430.Google Scholar
[2] Kendall, D. G. (1989) A survey of the statistical theory of shape (with discussion). Statist. Sci. 4, 87–120.Google Scholar
[3] Kendall, W. S. (1988) Symbolic computation and the diffusion of shapes of triads. Adv. Appl. Prob. 20, 775–797.Google Scholar
[4] Kent, J. T. (1978) Some probabilistic properties of Bessel functions. Ann. Prob. 6, 760–770.Google Scholar
[5] Mardia, K. V. (1989) Shape analysis of triangles through directional techniques. J. R. Statist. Soc. B 51, 449–458.Google Scholar
[6] Mardia, K. V. and Dryden, I. L. (1989) Shape distributions for landmark data. Adv. Appl. Prob. 21, 752–755.Google Scholar
[7] Molchanov, S. A. (1967) Martin boundaries for invariant Markov processes on a solvable group. Teor. Veroyatnost. 12, 310–314.Google Scholar
[8] Roberts, P. H. and Ursell, H. D. (1960) Random walk on a sphere and on a riemannian manifold. Phil. Trans. R. Soc. London A 252, 317–356.Google Scholar
[9] Rogers, L. C. G. and Williams, D. (1987) Diffusions, Markov Processes, and Martingales, Vol. 2. Wiley, Chichester.Google Scholar
[10] Watson, G. N. (1944) A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press.Google Scholar
[11] Yor, M. (1980) Loi de l'indice du lacet brownien, et distribution de Hartman-Watson. Z. Wahrscheinlichkeitsth. 53, 71–95.CrossRefGoogle Scholar