Abstract
The non isotropic noncentral elliptical shape distributions via pseudo-Wishart distribution are founded. This way, the classical shape theory is extended to non isotropic case and the normality assumption is replaced by assuming a elliptical distribution. In several cases, the new shape distributions are easily computable and then the inference procedure can be studied under exact densities. An application in Biology is studied under the classical gaussian approach and two non gaussian models.
Similar content being viewed by others
References
Bhattacharya, A. (2008). Statistical analysis on manifolds: A non-parametric approach for inference on shape spaces. Sankhya, Ser. A, Part 2, 70, 223–266.
Billingsley, P. (1986). Probability and Measure. John Wiley & Sons, New York.
Caro-Lopera, F.J., Díaz-García, J.A. and González-Farías, G. (2009). Noncentral elliptical configuration density. J. Multivariate Anal., 101, 32–43.
Davis, A.W. (1980). Invariant polynomials with two matrix arguments, extending the zonal polynomials. In Multivariate Analysis V, (P. R. Krishnaiah, ed.). North-Holland.
Díaz-García, J.A. and González-Farías, G. (2005). Singular random matrix decompositions: Distributions. J. Multivariate Anal., 194, 109–122.
Díaz-García, J.A. and Gutiérrez-Jáimez, R. (1997). Proof of the conjectures of H. Uhlig on the singular multivariate beta and the jacobian of a certain matrix transformation. Ann. Statist., 25, 2018–2023.
Díaz-García, J.A. and Gutiérrez-Jáimez, R. (2006). Wishart and Pseudo-Wishart distributions under elliptical laws and related distributions in the shape theory context. J. Statist. Plann. Inference, 136, 4176–4193.
Díaz-García, J.A., Gutiérrez-Jáimez, R. and Mardia, K.V. (1997). Wishart and Pseudo-Wishart distributions and some applications to shape theory. J. Multivariate Anal., 63, 73–87.
Dryden, I.L. and Mardia, K.V. (1989). Statistical shape analysis. John Wiley and Sons, Chichester.
Fang, K.T. and Zhang, Y.T. (1990). Generalized Multivariate Analysis. Science Press, Springer-Verlag, Beijing.
Goodall, C.G. (1991). Procustes methods in the statistical analysis of shape (with discussion). J. Roy. Statist. Soc. Ser. B, 53, 285–339.
Goodall, C.G. and Mardia, K.V. (1993). Multivariate Aspects of Shape Theory. Ann. Statist., 21, 848–866.
Gupta, A.K. and Varga, T. (1993). Elliptically Contoured Models in Statistics. Kluwer Academic Publishers, Dordrecht.
James, A.T. (1964). Distributions of matrix variate and latent roots derived from normal samples. Ann. Math. Statist., 35, 475–501.
Kass, R.E. and Raftery, A.E. (1995). Bayes factor. J. Amer. Statist. Soc., 90, 773–795.
Khatri, C.G. (1968). Some results for the singular normal multivariate regression models. Sankhyā A, 30, 267–280.
Koev, P. and Edelman, A. (2006). The efficient evaluation of the hypergeometric function of a matrix argument. Math. Comp., 75, 833–846.
Le, H.L. and Kendall, D.G. (1993). The Riemannian structure of Euclidean spaces: a novel environment for statistics. Ann. Statist., 21, 1225–1271.
Mardia, K.V. and Dryden, I.L. (1989). The Statistical Analysis of Shape Data. Biometrika, 76, 271–281
Muirhead, R.J. (1982). Aspects of multivariate statistical theory. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, Inc.
Raftery, A.E. (1995). Bayesian model selection in social research. Sociological Methodology, 25, 111–163.
Rao, C.R. (1973). Linear statistical inference and its applications (2nd ed.). John Wiley & Sons, New York.
Rissanen, J. (1978). Modelling by shortest data description. Automatica, 14, 465–471.
Uhlig, H. (1994). On singular Wishart and singular multivariate Beta distributions. Ann. Statist., 22, 395–405.
Yang, Ch.Ch. and Yang, Ch.Ch. (2007). Separating latent classes by information criteria. J. Classification, 24, 183–203.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Díaz-García, J.A., Caro-Lopera, F.J. Generalised Shape Theory Via Pseudo-Wishart Distribution. Sankhya A 75, 253–276 (2013). https://doi.org/10.1007/s13171-013-0024-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13171-013-0024-1
Keywords and phrases
- Shape theory
- maximum likelihood estimators
- zonal polynomials
- pseudo-Wishart distribution
- singular matrix multivariate distribution