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Generalised shape theory via SV decomposition I

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Abstract

This work finds in terms of zonal polynomials, the non isotropic noncentral elliptical shape distributions via singular value decomposition; it avoids the invariant polynomials and the open problems for their computation. The new shape distributions are easily computable and then the inference procedure is based on exact densities, instead of the published approximations and asymptotic distribution of isotropic models. An application of the technique is illustrated with a classical landmark data of Biology, for this, three Kotz type models are proposed (including Gaussian); then the best one is chosen by using a modified BIC criterion.

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References

  • Caro-Lopera FJ, Díaz-García JA, González-Farías G (2009) Noncentral elliptical configuration density. J Multivar Anal 101(1): 32–43

    Article  Google Scholar 

  • Davis AW (1980) Invariant polynomials with two matrix arguments, extending the zonal polynomials. In: Krishnaiah PR (ed.) Multivariate analysis V. North-Holland Publishing Company, Amsterdam, pp 287–299

    Google Scholar 

  • Díaz-García JA, Gutiérrez- Jáimez R, Mardia KV (1997) Wishart and Pseudo-Wishart distributions and some applications to shape theory. J Multivar Anal 63: 73–87

    Article  MATH  Google Scholar 

  • Díaz-García JA, Gutiérrez-Jáimez R, Ramos R (2003) Size-and-shape cone, shape disk and configuration densities for the elliptical models. Braz J Probab Stat 17: 135–146

    MathSciNet  MATH  Google Scholar 

  • Dryden IL, Mardia KV (1998) Statistical shape analysis. Wiley, Chichester

    MATH  Google Scholar 

  • Fang KT, Zhang YT (1990) Generalized multivariate analysis. Science Press, Springer, Beijing

    MATH  Google Scholar 

  • Goodall CR (1991) Procustes methods in the statistical analysis of shape (with discussion). J R Stat Soc Ser B 53: 285–339

    MathSciNet  MATH  Google Scholar 

  • Goodall CR, Mardia KV (1993) Multivariate aspects of shape theory. Ann Stat 21: 848–866

    Article  MathSciNet  MATH  Google Scholar 

  • Gupta AK, Varga T (1993) Elliptically contoured models in statistics. Kluwer, Dordrecht

    Book  MATH  Google Scholar 

  • James AT (1964) Distributions of matrix variate and latent roots derived from normal samples. Ann Math Stat 35: 475–501

    Article  MATH  Google Scholar 

  • Kass RE, Raftery AE (1995) Bayes factor. J Am Stat Soc 90: 773–795

    MATH  Google Scholar 

  • Koev P, Edelman A (2006) The efficient evaluation of the hypergeometric function of a matrix argument. Math Comput 75: 833–846

    Article  MathSciNet  MATH  Google Scholar 

  • Le HL, Kendall DG (1993) The Riemannian structure of Euclidean spaces: a novel environment for statistics. Ann Stat 21: 1225–1271

    Article  MathSciNet  MATH  Google Scholar 

  • Mardia KV, Dryden IL (1989) The statistical analysis of shape data. Biometrika 76(2): 271–281

    Article  MathSciNet  MATH  Google Scholar 

  • Muirhead RJ (1982) Aspects of multivariate statistical theory. Wiley series in probability and mathematical statistics. Wiley, New York

    Google Scholar 

  • Raftery AE (1995) Bayesian model selection in social research. Sociol Methodol 25: 111–163

    Article  Google Scholar 

  • Rissanen J (1978) Modelling by shortest data description. Automatica 14: 465–471

    Article  MATH  Google Scholar 

  • Yang ChCh, Yang ChCh (2007) Separating latent classes by information criteria. J Classif 24: 183–203

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Statistics and Computation, Universidad Autónoma Agraria Antonio Narro, 25315, Buenavista, Saltillo, Coahuila, Mexico

    José A. Díaz-García

  2. Department of Basic Sciences, Universidad de Medellín, Carrera 87 No.30–65, of., 5-103, Medellín, Colombia

    Francisco J. Caro-Lopera

Authors
  1. José A. Díaz-García
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  2. Francisco J. Caro-Lopera
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Correspondence to José A. Díaz-García.

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Díaz-García, J.A., Caro-Lopera, F.J. Generalised shape theory via SV decomposition I. Metrika 75, 541–565 (2012). https://doi.org/10.1007/s00184-010-0341-5

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  • DOI: https://doi.org/10.1007/s00184-010-0341-5

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