Estimation of mean form and mean form difference under elliptical laws

The matrix variate elliptical generalization of [30] is presented in this work. The published Gaussian case is revised and modified. Then, new aspects of identifiability and consistent estimation of mean form and mean form difference are considered under elliptical laws. For example, instead of usin...

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2017
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Universidad de Medellín
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Repositorio UDEM
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eng
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oai:repository.udem.edu.co:11407/4271
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http://hdl.handle.net/11407/4271
Palabra clave:
Coordinate free approach
Matrix variate elliptical distribution
Matrix variate Gaussian distribution
Non-central singular Pseudo-Wishart distribution
Statistical shape theory
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oai_identifier_str oai:repository.udem.edu.co:11407/4271
network_acronym_str REPOUDEM2
network_name_str Repositorio UDEM
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dc.title.spa.fl_str_mv Estimation of mean form and mean form difference under elliptical laws
title Estimation of mean form and mean form difference under elliptical laws
spellingShingle Estimation of mean form and mean form difference under elliptical laws
Coordinate free approach
Matrix variate elliptical distribution
Matrix variate Gaussian distribution
Non-central singular Pseudo-Wishart distribution
Statistical shape theory
title_short Estimation of mean form and mean form difference under elliptical laws
title_full Estimation of mean form and mean form difference under elliptical laws
title_fullStr Estimation of mean form and mean form difference under elliptical laws
title_full_unstemmed Estimation of mean form and mean form difference under elliptical laws
title_sort Estimation of mean form and mean form difference under elliptical laws
dc.contributor.affiliation.spa.fl_str_mv Díaz-García, J.A., Universidad Autónoma Agraria Antonio Narro, Calzada Antonio Narro 1923, Col. Buenavista, Saltillo, Coahuila, Mexico
Caro-Lopera, F.J., Faculty of Basic Sciences, Universidad de Medellín, Medellín, Colombia
dc.subject.keyword.eng.fl_str_mv Coordinate free approach
Matrix variate elliptical distribution
Matrix variate Gaussian distribution
Non-central singular Pseudo-Wishart distribution
Statistical shape theory
topic Coordinate free approach
Matrix variate elliptical distribution
Matrix variate Gaussian distribution
Non-central singular Pseudo-Wishart distribution
Statistical shape theory
description The matrix variate elliptical generalization of [30] is presented in this work. The published Gaussian case is revised and modified. Then, new aspects of identifiability and consistent estimation of mean form and mean form difference are considered under elliptical laws. For example, instead of using the Euclidean distance matrix for the consistent estimates, exact formulae are derived for the moments of the matrix B = Xc(Xc)T; where Xcis the centered landmark matrix. Finally, a complete application in Biology is provided; it includes estimation, model selection and hypothesis testing. © 2017, Institute of Mathematical Statistics. All rights reserved.
publishDate 2017
dc.date.accessioned.none.fl_str_mv 2017-12-19T19:36:43Z
dc.date.available.none.fl_str_mv 2017-12-19T19:36:43Z
dc.date.created.none.fl_str_mv 2017
dc.type.eng.fl_str_mv Article
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dc.identifier.issn.none.fl_str_mv 19357524
dc.identifier.uri.none.fl_str_mv http://hdl.handle.net/11407/4271
dc.identifier.doi.none.fl_str_mv 10.1214/17-EJS1289
dc.identifier.reponame.spa.fl_str_mv reponame:Repositorio Institucional Universidad de Medellín
dc.identifier.instname.spa.fl_str_mv instname:Universidad de Medellín
identifier_str_mv 19357524
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reponame:Repositorio Institucional Universidad de Medellín
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url http://hdl.handle.net/11407/4271
dc.language.iso.none.fl_str_mv eng
language eng
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dc.relation.ispartofes.spa.fl_str_mv Electronic Journal of Statistics
Electronic Journal of Statistics Volume 11, Issue 1, 2017, Pages 2424-2460
dc.relation.references.spa.fl_str_mv Arnold, S. F. (1981). The Theory of Linear Models and Multivariate Analysis.
Bookstein, F. L. (1986). Size and shape spaces for landmark data in two dimensions. Statistical Science, 1(2), 181-222. doi:10.1214/ss/1177013696
Caro-Lopera, F. J., & Díaz-García, J. A. (2012). Matrix kummer-pearson VII relation and polynomial pearson VII configuration density. Journal of the Iranian Statistical Society, 11(2), 217-230.
Caro-Lopera, F. J., Díaz-García, J. A., & González-Farías, G. (2014). Inference in affine shape theory under elliptical models. Journal of the Korean Statistical Society, 43(1), 67-77. doi:10.1016/j.jkss.2013.05.004
Caro-Lopera, F. J., Díaz-García, J. A., & González-Farías, G. (2010). Noncentral elliptical configuration density. Journal of Multivariate Analysis, 101(1), 32-43. doi:10.1016/j.jmva.2009.03.004
Caro-Lopera, F. J., Leiva, V., & Balakrishnan, N. (2012). Connection between the hadamard and matrix products with an application to matrix-variate birnbaum-saunders distributions. Journal of Multivariate Analysis, 104(1), 126-139. doi:10.1016/j.jmva.2011.07.004
Díaz-García, J. A. (1994). Contributions to the Theory of Wishart and Multivariate Elliptical Distributions.
Díaz-García, J. A., & Caro-Lopera, F. J. (2013). Generalised shape theory via pseudo-wishart distribution. Sankhya: The Indian Journal of Statistics, 75 A(PART2), 253-276.
Díaz-García, J. A., & Caro-Lopera, F. J. (2012). Generalised shape theory via SV decomposition I. Metrika, 75(4), 541-565. doi:10.1007/s00184-010-0341-5
Díaz-García, J. A., & Caro-Lopera, F. J. (2012). Statistical theory of shape under elliptical models and singular value decompositions. Journal of Multivariate Analysis, 103(1), 77-92. doi:10.1016/j.jmva.2011.06.010
Díaz-García, J. A., & Caro-Lopera, F. J. (2014). Statistical theory of shape under elliptical models via QR decomposition. Statistics, 48(2), 456-472. doi:10.1080/02331888.2013.801973
Díaz-García, J. A., & González-Farías, G. (2005). Singular random matrix decompositions: Distributions. Journal of Multivariate Analysis, 94(1), 109-122. doi:10.1016/j.jmva.2004.08.003
Díaz-García, J. A., & Gutiérrez Jáimez, R. (1996). Matrix differential calculus and moments of a random matrix elliptical. Serie Colecci´on“Estadística Multivariable Y Procesos Estocásticos.
Díaz-García, J. A., Gutiérrez-Jáimez, R., & Ramos-Quiroga, R. (2003). Size-and-shape cone, shape disk and configuration densities for the elliptical models. Braz.J.Probab.Stat., 17, 135-146.
Díaz-García, J. A., & Jáimez, R. G. (2006). Wishart and pseudo-wishart distributions under elliptical laws and related distributions in the shape theory context. Journal of Statistical Planning and Inference, 136(12), 4176-4193. doi:10.1016/j.jspi.2005.08.045
Dryden, I. L., Koloydenko, A., & Zhou, D. (2009). Non-euclidean statistics for covariance matrices, with applications to diffusion tensor imaging. Annals of Applied Statistics, 3(3), 1102-1123. doi:10.1214/09-AOAS249
Dryden, I. L., & Mardia, K. V. (1998). Statistical Shape Analysis.
Dutilleuel, P. (1999). The MLE algorithm for the matrix normal distribution. Journal of Statistical Computation and Simulation, 64(2), 105-123.
Fang, K. T., Kotz, S., & Ng, K. W. (1990). Symmetric multivariate and related distributions. Symmetric Multivariate and Related Distributions.
Fang, K. T., & Zhang, Y. T. (1990). Generalized multivariate analysis. Generalized Multivariate Analysis.
Goodall, C. (1991). Procrustes methods in the statistical analysis of shape. J.R.Statist.Soc.B, 53(2), 285-339.
Goodall, C. R., & Mardia, K. V. (1993). Multivariate aspects of shape theory. Ann.Statist., 21, 848-866.
Gupta, A. K., & Varga, T. (1993). Elliptically Contoured Models in Statistics.
Kendall, D. G. (1984). Shape manifolds, procrustean metrics, and complex projective spaces. Bulletin of the London Mathematical Society, 16(2), 81-121. doi:10.1112/blms/16.2.81
Kent, J. T. (1992). New directions in shape analysis. The Art of Statistical Science, 115-127.
Khatri, C. G. (1968). Some results for the singular normal multivariate regression models. Sankhyā A, 30, 267-280.
Koev, P., & Edelman, A. (2006). The efficient evaluation of the hypergeometric function of a matrix argument. Mathematics of Computation, 75(254), 833-846. doi:10.1090/S0025-5718-06-01824-2
Le, H., & Kendall, D. G. (1993). The riemannian structure of euclidean shape spaces: A novel environment for statistics. Annals of Statistics, 21(3), 1225-1271.
Lele, S. (1993). Euclidean distance matrix analysis (EDMA): Estimation of mean form and mean form difference. Mathematical Geology, 25(5), 573-602. doi:10.1007/BF00890247
Lele, S. (1991). Some comments on coordinate‐free and scale‐invariant methods in morphometrics. American Journal of Physical Anthropology, 85(4), 407-417. doi:10.1002/ajpa.1330850405
Lele, S., & Richtsmeier, J. T. (1991). Euclidean distance matrix analysis: A coordinate‐free approach for comparing biological shapes using landmark data. American Journal of Physical Anthropology, 86(3), 415-427. doi:10.1002/ajpa.1330860307
Lele, S., & Richtsmeier, J. T. (1990). Statistical models in morphometrics: Are they realistic? Systematic Zoology, 39(1), 60-69. doi:10.2307/2992208
Magnus, J. R., & Neudecker, H. (1979). The commutation matrix: Some properties and applications. Ann.Statist., 7(2), 381-394.
Mardia, K. V., & Dryden, I. L. (1989). The statistical analysis of shape data. Biometrika, 76(2), 271-281. doi:10.1093/biomet/76.2.271
Mood, A. M., Graybill, F. A., & Boes, D. C. (1974). Introduction to the Theory of Statistics.
Muirhead, R. J. (1982). "Aspects of multivariate statistical theory". Aspects of Multivariate Statistical Theory.
Nadarajah, S. (2003). The kotz-type distribution with applications. Statistics, 37(4), 341-358. doi:10.1080/0233188031000078060
Rao, C. R. (1973). Linear Statistical Inference and its Applications.
Richtsmeier, J. T., DeLeon, V. B., & Lele, S. R. (2002). The promise of geometric morphometrics. Yearbook of Physical Anthropology, 45, 63-91. doi:10.1002/ajpa.10174
Sawyer, P. (1997). Spherical functions on symmetric cones. Transactions of the American Mathematical Society, 349(9), 3569-3584.
Walker, J. A. (2000). Ability of geometric morphometric methods to estimate a known covariance matrix. Systematic Biology, 49(4), 686-696.
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spelling 2017-12-19T19:36:43Z2017-12-19T19:36:43Z201719357524http://hdl.handle.net/11407/427110.1214/17-EJS1289reponame:Repositorio Institucional Universidad de Medellíninstname:Universidad de MedellínThe matrix variate elliptical generalization of [30] is presented in this work. The published Gaussian case is revised and modified. Then, new aspects of identifiability and consistent estimation of mean form and mean form difference are considered under elliptical laws. For example, instead of using the Euclidean distance matrix for the consistent estimates, exact formulae are derived for the moments of the matrix B = Xc(Xc)T; where Xcis the centered landmark matrix. Finally, a complete application in Biology is provided; it includes estimation, model selection and hypothesis testing. © 2017, Institute of Mathematical Statistics. All rights reserved.engInstitute of Mathematical StatisticsFacultad de Ciencias Básicashttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85020382799&doi=10.1214%2f17-EJS1289&partnerID=40&md5=8e564bfa96c8a13ee5eafe6ebc225c38Electronic Journal of StatisticsElectronic Journal of Statistics Volume 11, Issue 1, 2017, Pages 2424-2460Arnold, S. F. (1981). The Theory of Linear Models and Multivariate Analysis.Bookstein, F. L. (1986). Size and shape spaces for landmark data in two dimensions. Statistical Science, 1(2), 181-222. doi:10.1214/ss/1177013696Caro-Lopera, F. J., & Díaz-García, J. A. (2012). Matrix kummer-pearson VII relation and polynomial pearson VII configuration density. Journal of the Iranian Statistical Society, 11(2), 217-230.Caro-Lopera, F. J., Díaz-García, J. A., & González-Farías, G. (2014). Inference in affine shape theory under elliptical models. Journal of the Korean Statistical Society, 43(1), 67-77. doi:10.1016/j.jkss.2013.05.004Caro-Lopera, F. J., Díaz-García, J. A., & González-Farías, G. (2010). Noncentral elliptical configuration density. Journal of Multivariate Analysis, 101(1), 32-43. doi:10.1016/j.jmva.2009.03.004Caro-Lopera, F. J., Leiva, V., & Balakrishnan, N. (2012). Connection between the hadamard and matrix products with an application to matrix-variate birnbaum-saunders distributions. Journal of Multivariate Analysis, 104(1), 126-139. doi:10.1016/j.jmva.2011.07.004Díaz-García, J. A. (1994). Contributions to the Theory of Wishart and Multivariate Elliptical Distributions.Díaz-García, J. A., & Caro-Lopera, F. J. (2013). Generalised shape theory via pseudo-wishart distribution. Sankhya: The Indian Journal of Statistics, 75 A(PART2), 253-276.Díaz-García, J. A., & Caro-Lopera, F. J. (2012). Generalised shape theory via SV decomposition I. Metrika, 75(4), 541-565. doi:10.1007/s00184-010-0341-5Díaz-García, J. A., & Caro-Lopera, F. J. (2012). Statistical theory of shape under elliptical models and singular value decompositions. Journal of Multivariate Analysis, 103(1), 77-92. doi:10.1016/j.jmva.2011.06.010Díaz-García, J. A., & Caro-Lopera, F. J. (2014). Statistical theory of shape under elliptical models via QR decomposition. Statistics, 48(2), 456-472. doi:10.1080/02331888.2013.801973Díaz-García, J. A., & González-Farías, G. (2005). Singular random matrix decompositions: Distributions. Journal of Multivariate Analysis, 94(1), 109-122. doi:10.1016/j.jmva.2004.08.003Díaz-García, J. A., & Gutiérrez Jáimez, R. (1996). Matrix differential calculus and moments of a random matrix elliptical. Serie Colecci´on“Estadística Multivariable Y Procesos Estocásticos.Díaz-García, J. A., Gutiérrez-Jáimez, R., & Ramos-Quiroga, R. (2003). Size-and-shape cone, shape disk and configuration densities for the elliptical models. Braz.J.Probab.Stat., 17, 135-146.Díaz-García, J. A., & Jáimez, R. G. (2006). Wishart and pseudo-wishart distributions under elliptical laws and related distributions in the shape theory context. Journal of Statistical Planning and Inference, 136(12), 4176-4193. doi:10.1016/j.jspi.2005.08.045Dryden, I. L., Koloydenko, A., & Zhou, D. (2009). Non-euclidean statistics for covariance matrices, with applications to diffusion tensor imaging. Annals of Applied Statistics, 3(3), 1102-1123. doi:10.1214/09-AOAS249Dryden, I. L., & Mardia, K. V. (1998). Statistical Shape Analysis.Dutilleuel, P. (1999). The MLE algorithm for the matrix normal distribution. Journal of Statistical Computation and Simulation, 64(2), 105-123.Fang, K. T., Kotz, S., & Ng, K. W. (1990). Symmetric multivariate and related distributions. Symmetric Multivariate and Related Distributions.Fang, K. T., & Zhang, Y. T. (1990). Generalized multivariate analysis. Generalized Multivariate Analysis.Goodall, C. (1991). Procrustes methods in the statistical analysis of shape. J.R.Statist.Soc.B, 53(2), 285-339.Goodall, C. R., & Mardia, K. V. (1993). Multivariate aspects of shape theory. Ann.Statist., 21, 848-866.Gupta, A. K., & Varga, T. (1993). Elliptically Contoured Models in Statistics.Kendall, D. G. (1984). Shape manifolds, procrustean metrics, and complex projective spaces. Bulletin of the London Mathematical Society, 16(2), 81-121. doi:10.1112/blms/16.2.81Kent, J. T. (1992). New directions in shape analysis. The Art of Statistical Science, 115-127.Khatri, C. G. (1968). Some results for the singular normal multivariate regression models. Sankhyā A, 30, 267-280.Koev, P., & Edelman, A. (2006). The efficient evaluation of the hypergeometric function of a matrix argument. Mathematics of Computation, 75(254), 833-846. doi:10.1090/S0025-5718-06-01824-2Le, H., & Kendall, D. G. (1993). The riemannian structure of euclidean shape spaces: A novel environment for statistics. Annals of Statistics, 21(3), 1225-1271.Lele, S. (1993). Euclidean distance matrix analysis (EDMA): Estimation of mean form and mean form difference. Mathematical Geology, 25(5), 573-602. doi:10.1007/BF00890247Lele, S. (1991). Some comments on coordinate‐free and scale‐invariant methods in morphometrics. American Journal of Physical Anthropology, 85(4), 407-417. doi:10.1002/ajpa.1330850405Lele, S., & Richtsmeier, J. T. (1991). Euclidean distance matrix analysis: A coordinate‐free approach for comparing biological shapes using landmark data. American Journal of Physical Anthropology, 86(3), 415-427. doi:10.1002/ajpa.1330860307Lele, S., & Richtsmeier, J. T. (1990). Statistical models in morphometrics: Are they realistic? Systematic Zoology, 39(1), 60-69. doi:10.2307/2992208Magnus, J. R., & Neudecker, H. (1979). The commutation matrix: Some properties and applications. Ann.Statist., 7(2), 381-394.Mardia, K. V., & Dryden, I. L. (1989). The statistical analysis of shape data. Biometrika, 76(2), 271-281. doi:10.1093/biomet/76.2.271Mood, A. M., Graybill, F. A., & Boes, D. C. (1974). Introduction to the Theory of Statistics.Muirhead, R. J. (1982). "Aspects of multivariate statistical theory". Aspects of Multivariate Statistical Theory.Nadarajah, S. (2003). The kotz-type distribution with applications. Statistics, 37(4), 341-358. doi:10.1080/0233188031000078060Rao, C. R. (1973). Linear Statistical Inference and its Applications.Richtsmeier, J. T., DeLeon, V. B., & Lele, S. R. (2002). The promise of geometric morphometrics. Yearbook of Physical Anthropology, 45, 63-91. doi:10.1002/ajpa.10174Sawyer, P. (1997). Spherical functions on symmetric cones. Transactions of the American Mathematical Society, 349(9), 3569-3584.Walker, J. A. (2000). Ability of geometric morphometric methods to estimate a known covariance matrix. Systematic Biology, 49(4), 686-696.ScopusEstimation of mean form and mean form difference under elliptical lawsArticleinfo:eu-repo/semantics/articlehttp://purl.org/coar/version/c_970fb48d4fbd8a85http://purl.org/coar/resource_type/c_6501http://purl.org/coar/resource_type/c_2df8fbb1Díaz-García, J.A., Universidad Autónoma Agraria Antonio Narro, Calzada Antonio Narro 1923, Col. Buenavista, Saltillo, Coahuila, MexicoCaro-Lopera, F.J., Faculty of Basic Sciences, Universidad de Medellín, Medellín, ColombiaDíaz-García J.A.Caro-Lopera F.J.Universidad Autónoma Agraria Antonio Narro, Calzada Antonio Narro 1923, Col. Buenavista, Saltillo, Coahuila, MexicoFaculty of Basic Sciences, Universidad de Medellín, Medellín, ColombiaCoordinate free approachMatrix variate elliptical distributionMatrix variate Gaussian distributionNon-central singular Pseudo-Wishart distributionStatistical shape theoryThe matrix variate elliptical generalization of [30] is presented in this work. The published Gaussian case is revised and modified. Then, new aspects of identifiability and consistent estimation of mean form and mean form difference are considered under elliptical laws. For example, instead of using the Euclidean distance matrix for the consistent estimates, exact formulae are derived for the moments of the matrix B = Xc(Xc)T; where Xcis the centered landmark matrix. Finally, a complete application in Biology is provided; it includes estimation, model selection and hypothesis testing. © 2017, Institute of Mathematical Statistics. All rights reserved.http://purl.org/coar/access_right/c_16ecTHUMBNAIL33. Estimation of mean form and mean form difference under elliptical laws.pdf.jpg33. 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