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Shape averages and their Bias

Part of: Multivariate analysis

Published online by Cambridge University Press:  01 July 2016

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

The paper considers the bias of Bookstein's mean estimator for shape under the isotropic normal model. This work depends on certain distributional properties of shape variables. An alternative unbiased modified estimator is proposed and its performance is compared with various estimators, including Procrustes and the exact maximum likelihood estimator, in a simulation study.

Keywords

LANDMARKSPROCRUSTESSHAPESIZETRIANGLE

MSC classification

Primary: 62H11: Directional data; spatial statistics

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 26 , Issue 2 , June 1994 , pp. 334 - 340
Copyright
Copyright © Applied Probability Trust 1994 

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References

Bookstein, F. L. (1986) Size and shape spaces for landmark data in two dimensions (with discussion). Statist. Sci. 4, 181–242.Google Scholar
Bookstein, F. L. (1991) Morphometric Tools for Landmark Data. Cambridge University Press.Google Scholar
Bookstein, F. L. and Sampson, P. D. (1990) Statistical models for geometric components of shape change. Commun. Statist. Theory Meth. 19, 1939–1972.Google Scholar
Dryden, I. L. (1989) The statistical analysis of shape data. PhD Dissertation, University of Leeds.Google Scholar
Dryden, I. L. (1991) Discussion to Goodall (1991). J. R. Statist. Soc. B 53, 327–328.Google Scholar
Goodall, C. R. (1991) Procrustes methods in the statistical analysis of shape. J. R. Statist. Soc. B 53, 285–339.Google Scholar
Goodall, C. R. (1993) Shape as a dependent variable in statistical models. Presentation IMS Philadelphia meeting 1993. Abstract: IMS Bull. 22, 91.Google Scholar
Gower, J. C. (1975) Generalized Procrustes analysis. Psychometrika 40, 33–50.Google Scholar
Kendall, D. G. (1984) Shape manifolds, procrustean metrics and complex projective spaces. Bull. London Math. Soc. 16, 81–121.Google Scholar
Kent, J. T. (1991) Discussion to Goodall (1991). J. R. Statist. Soc. B 53, 324–325.Google Scholar
Mardia, K. V. (1972) Statistics of Directional Data. Academic Press, London.Google Scholar
Mardia, K. V. (1989) Shape analysis of triangles through directional techniques. J. R. Statist. Soc. B 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) Statistical analysis of shape data. Biometrika 76, 271–281.CrossRefGoogle Scholar
Stoyan, D. (1990) Estimation of distances and variances in Bookstein's landmark model. Biom. J. 32, 843–849.Google Scholar
Stoyan, D. and Frenz, M. (1993) Estimating mean landmark triangles. Biom. J. 35, 643–647.Google Scholar
Ten Berge, J. M. F. (1977) Orthogonal Procrustes rotation for two or more matrices. Psychometrika 42, 267–276.Google Scholar
Watson, G. N. (1944) Theory of Bessel Functions, 2nd edn. Cambridge University Press.Google Scholar
Ziezold, H. (1989) On expected figures in the plane. In Geobild '89, ed. Hubler, A., Nagel, W., Ripley, B. D., and Werner, G. pp. 105–110. Akademie-Verlag, Berlin.Google Scholar