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General shape distributions in a plane

Published online by Cambridge University Press:  01 July 2016

I. L. Dryden  and
K. V. Mardia
I. L. Dryden
Affiliation:
University of Leeds
K. V. Mardia*
Affiliation:
University of Leeds
*
∗Postal address: Department of Statistics, University of Leeds, Leeds, LS2 9JT, UK.
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Abstract

In this paper we investigate the exact shape distribution for general Gaussian labelled point configurations in two dimensions. The shape density is written in a closed form, in terms of Kendall's or Bookstein's shape variables. The distribution simplifies considerably in certain cases, including the complex normal, isotropic, circular Markov and equal means cases. Various asymptotic properties of the distribution are investigated, including a large variation distribution and the normal approximation for small variations. The triangle case is considered in particular detail, and we compare the density with simulated densities for some examples. Finally, we consider inference problems, with an application in biology.

Keywords

COMPLEX NORMALCYCLIC MARKOV PROCESSKENDALL SHAPE SPACELAGUERRE POLYNOMIALLEGENDRE POLYNOMIALPROCRUSTEAN DISTANCESHAPETRIANGLE
Type
Research Article
Information
Advances in Applied Probability , Volume 23 , Issue 2 , June 1991 , pp. 259 - 276
Copyright
Copyright © Applied Probability Trust 1991 

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References

Bookstein, F. L. (1986) Size and shape spaces for landmark data in two dimensions. Statist. Sci. 1, 181–242.Google Scholar
Kendall, D. G. (1984) Shape manifolds, procrustean metrics and complex projective spaces. Bull. London Math. Soc. 16, 81–121.Google Scholar
Kendall, D. G. (1986) In discussion to Bookstein (1986). Statist. Sci. 1, 222–226.Google Scholar
Kendall, D. G. (1989) A survey of the statistical theory of shape. Statist. Sci. 4, 87–120.Google Scholar
Mardia, K. V. (1989) Shape analysis of triangles through directional techniques. J. R. Statist. Soc. 51, 449–458.Google Scholar
Mardia, K. V. and Dryden, I. L. (1989a) Shape distributions for landmark data. Adv. Appl. Prob. 21, 742–755.Google Scholar
Mardia, K. V. and Dryden, I. L. (1989b) The statistical analysis of shape data. Biometrika 76, 271–281.Google Scholar
Rainville, E. D. (1960) Special Functions. Macmillan, New York.Google Scholar