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Matrix variate Birnbaum-Saunders distribution under elliptical models. (English) Zbl 1441.62136
Summary: This paper derives the elliptical matrix variate version of the well known univariate Birnbaum-Saunders distribution [Z. W. Birnbaum and S. C. Saunders, J. Appl. Probab. 6, 319–327 (1969; Zbl 0209.49801)]. A generalisation based on a matrix transformation is proposed, instead of the independent element-to-element elliptical extension of the Gaussian univariate case. Some results on Jacobians were needed to derive the new matrix variate distribution. A number of particular distributions are studied and some basic properties are found. Finally, an example based on real data of two populations is given and the maximum likelihood estimates are obtained for the class of Kotz models. A comparison with the Gaussian kernel is also given by using a modified BIC criterion.
62H10 Multivariate distribution of statistics
62N05 Reliability and life testing
62E15 Exact distribution theory in statistics
60E05 Probability distributions: general theory
15A23 Factorization of matrices
15A09 Theory of matrix inversion and generalized inverses
15B52 Random matrices (algebraic aspects)
Full Text: DOI
[1] Balakrishnan, N.; Kundu, D., Birnbaum-Saunders distribution: A review of model, analysis and applications, Appl. Stoch. Models Bus. Ind., 35, 1, 1-151 (2019), with discussion · Zbl 1428.62072
[2] Birnbaum, Z. W.; Saunders, S. C., A new family of life distributions, J. Appl. Probab., 6, 637-652 (1969) · Zbl 0162.22303
[3] Cadet, A., Polar coordinates in \(\mathbf{R}^{n p} \); application to the computation of the Wishart and beta laws, Sankhyā A, 58, 101-113 (1996) · Zbl 0904.62064
[4] Caro-Lopera, F. J.; Díaz-García, J. A., Diagonalization matrix and its application in distribution theory, Statistics, 50, 4, 870-880 (2016) · Zbl 1356.62022
[5] Caro-Lopera, F. J.; Leiva, V.; Balakrishnan, N., Connection between the hadamard and matrix products with an application to matrix-variate Birnbaum-Saunders distributions, J. Multivariate Anal., 104, 1, 126-139 (2012) · Zbl 1298.15017
[6] Chen, J. J.; Novick, M. R., Bayesian analysis for binomial models with generalized beta prior distributions, J. Educ. Stat., 9, 163-175 (1984)
[7] Davis, A. W., Invariant polynomials with two matrix arguments, extending the zonal polynomials: Applications to multivariate distribution theory, Ann. Inst. Stat. Math. A, 31, 465-485 (1979) · Zbl 0463.62045
[8] Desmond, A., Stochastic models of failure in random enviorments, Canad. J. Stat., 13, 171-183 (1985) · Zbl 0581.60073
[9] Díaz-García, J. A.; Caro-Lopera, F. J.; Pérez Ramírez, F. O., Multivector variate distributions, Sankhyā (2019), (in press)
[10] Díaz-García, J. A.; Domínguez Molina, J. R., Some generalisations of Birnbaum-Saunders and sinh-normal distributions, Int. Math. Forum., 1, 35, 1709-1727 (2006) · Zbl 1109.62036
[11] Díaz-García, J. A.; Domínguez Molina, J. R., A new family of life distributions for dependent data: Estimation, Comput. Statist. Data Anal., 51, 12, 5927-5939 (2007) · Zbl 1445.62110
[12] Díaz-García, J. A.; Gutiérrez-Jáimez, R., Functions of singular random matrices and its applications, Test, 14, 2, 475-487 (2005) · Zbl 1087.62061
[13] Díaz-García, J. A.; Leiva-Sánchez, V., A new family of life distributions based on elliptically contoured distributions, J. Stat. Plan. Inf., 128, 2, 445-457 (2005) · Zbl 1087.62011
[14] Díaz-García, J. A.; Leiva-Sánchez, V., A new family of life distributions based on the elliptically contoured distributions, J. Stat. Plan. Inf.. J. Stat. Plan. Inf., J. Statist. Plann. Inference, 128, 2, 445-457 (2005), Erratum to · Zbl 1087.62011
[15] Fang, K. T.; Zhang, Y. T., Generalized Multivariate Analysis (1990), Science Press, Springer-Verlag: Science Press, Springer-Verlag Beijing
[16] Gupta, A. K.; Varga, Y.; Bodnar, T., Elliptical Contoured Models in Statistics and Portfolio Theory (2013), Springer: Springer New York · Zbl 1306.62028
[17] Herz, C. S., Bessel functions of matrix argument, Ann. of Math., 61, 3, 474-523 (1955) · Zbl 0066.32002
[18] James, A. T., Normal multivariate analysis and the orthogonal group, Ann. Math. Stat., 25, 1, 40-75 (1954) · Zbl 0055.13203
[19] Kass, R. E.; Raftery, A. E., Bayes factor, J. Am. Stat. Soc., 90, 773-795 (1995) · Zbl 0846.62028
[20] Kotz, S.; Nadarajah, S., Multivariate \(T\) Distributions and their Applications (2004), Cambridge University Press: Cambridge University Press United Kingdom · Zbl 1100.62059
[21] Libby, D. L.; Novick, M. R., Multivariate generalized beta distributions with applications to utility assessment, J. Educ. Stat., 7, 271-294 (1982)
[22] Magnus, J. R., Linear Structures (1988), Charles Griffin & Company Ltd: Charles Griffin & Company Ltd London · Zbl 0667.15010
[23] Magnus, J. R.; Neudecker, H., Matrix Differential Calculus with Application in Statistics and Econometrics (2007), John Wiley & Sons: John Wiley & Sons Chichester
[24] Mathai, A. M., Jacobian of Matrix Transformations and Functions of Matrix Argument (1997), World Scinentific: World Scinentific Singapore · Zbl 0889.33001
[25] Muirhead, R. J., Aspects of Multivariate Statistical Theory (2005), John Wiley & Sons: John Wiley & Sons New York
[26] Ng, H. K.T.; Kundu, D.; Balakrishnan, N., Modified moment estimation for the two-parameter Birnbaum-Saunders distribution, Comput. Stat. Data Anal., 43, 2003, 283-298 (2003) · Zbl 1429.62451
[27] Olkin, I.; Rubin, H., Multivariate beta distributions and independence properties of Wishart distribution, Ann. Math. Stat., 37, 1, 297-269 (1966), Correction · Zbl 0128.14002
[28] Raftery, A. E., Bayesian model selection in social research, Sociol. Methodol., 25, 111-163 (1995)
[29] Rao, C. R., Linear Statistical Inference and Its Applications (2005), John Wiley & Sons: John Wiley & Sons New York
[30] Roy, S. N., Some Aspects of Multivariate Analysis (1957), John Wiley & Sons, Inc.: John Wiley & Sons, Inc. New York
[31] Sánchez, L.; Leiva, V.; Caro-Lopera, F.; Cysneiros, F. J., On matrix-variate BirnbaumSaunders distributions and their estimation and application, Braz. J. Probab. Stat., 29, 4, 790-812 (2015) · Zbl 1329.60013
[32] Srivastava, M. S.; Khatri, C. G., An Introduction To Multivariate Analysis (1979), North-Holland Publ: North-Holland Publ Amsterdam · Zbl 0421.62034
[33] Yang, Ch. Ch.; Yang, Ch. Ch., Separating latent classes by information criteria, J Classification, 24, 183-203 (2007) · Zbl 1234.62102
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