Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T15:08:53.706Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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

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

[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