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Generalized multivariate Birnbaum–Saunders distributions and related inferential issues

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  • Kundu, Debasis
  • Balakrishnan, N.
  • Jamalizadeh, Ahad

Abstract

Birnbaum and Saunders introduced in 1969 a two-parameter lifetime distribution which has been used quite successfully to model a wide variety of univariate positively skewed data. Diaz-Garcia and Leiva-Sanchez [8] proposed a generalized Birnbaum–Saunders distribution by using an elliptically symmetric distribution in place of the normal distribution. Recently, Kundu et al. [13] introduced a bivariate Birnbaum–Saunders distribution, based on a transformation of a bivariate normal distribution, and discussed its properties and associated inferential issues. In this paper, we construct a generalized multivariate Birnbaum–Saunders distribution, by using the multivariate elliptically symmetric distribution as a base kernel for the transformation instead of the multivariate normal distribution. Different properties of this distribution are obtained in the general case. Special emphasis is placed on statistical inference for two particular cases: (i) multivariate normal kernel and (ii) multivariate-t kernels. We use the maximized log-likelihood values for selecting the best kernel function. Finally, a data analysis is presented for illustrative purposes.

Suggested Citation

  • Kundu, Debasis & Balakrishnan, N. & Jamalizadeh, Ahad, 2013. "Generalized multivariate Birnbaum–Saunders distributions and related inferential issues," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 230-244.
    • Handle: RePEc:eee:jmvana:v:116:y:2013:i:c:p:230-244
    DOI: 10.1016/j.jmva.2012.10.017
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    1. Vilca, Filidor & Santana, Lucia & Leiva, Víctor & Balakrishnan, N., 2011. "Estimation of extreme percentiles in Birnbaum-Saunders distributions," Computational Statistics & Data Analysis, Elsevier, vol. 55(4), pages 1665-1678, April.
    2. Sampson, Allan R., 1983. "Positive dependence properties of elliptically symmetric distributions," Journal of Multivariate Analysis, Elsevier, vol. 13(2), pages 375-381, June.
    3. Leiva, Victor & Riquelme, Marco & Balakrishnan, N. & Sanhueza, Antonio, 2008. "Lifetime analysis based on the generalized Birnbaum-Saunders distribution," Computational Statistics & Data Analysis, Elsevier, vol. 52(4), pages 2079-2097, January.
    4. Kundu, Debasis & Kannan, Nandini & Balakrishnan, N., 2008. "On the hazard function of Birnbaum-Saunders distribution and associated inference," Computational Statistics & Data Analysis, Elsevier, vol. 52(5), pages 2692-2702, January.
    5. Cambanis, Stamatis & Huang, Steel & Simons, Gordon, 1981. "On the theory of elliptically contoured distributions," Journal of Multivariate Analysis, Elsevier, vol. 11(3), pages 368-385, September.
    6. Azevedo, Cecilia & Leiva, Víctor & Athayde, Emilia & Balakrishnan, N., 2012. "Shape and change point analyses of the Birnbaum–Saunders-t hazard rate and associated estimation," Computational Statistics & Data Analysis, Elsevier, vol. 56(12), pages 3887-3897.
    7. Karlin, Samuel & Rinott, Yosef, 1980. "Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions," Journal of Multivariate Analysis, Elsevier, vol. 10(4), pages 467-498, December.
    8. Kundu, Debasis & Balakrishnan, N. & Jamalizadeh, A., 2010. "Bivariate Birnbaum-Saunders distribution and associated inference," Journal of Multivariate Analysis, Elsevier, vol. 101(1), pages 113-125, January.
    9. Ng, H. K. T. & Kundu, D. & Balakrishnan, N., 2003. "Modified moment estimation for the two-parameter Birnbaum-Saunders distribution," Computational Statistics & Data Analysis, Elsevier, vol. 43(3), pages 283-298, July.
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    Cited by:

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    2. Luis Sánchez & Víctor Leiva & Manuel Galea & Helton Saulo, 2020. "Birnbaum-Saunders Quantile Regression Models with Application to Spatial Data," Mathematics, MDPI, vol. 8(6), pages 1-17, June.
    3. Kakizawa, Yoshihide, 2022. "Multivariate elliptical-based Birnbaum–Saunders kernel density estimation for nonnegative data," Journal of Multivariate Analysis, Elsevier, vol. 187(C).
    4. Vilca, Filidor & Balakrishnan, N. & Zeller, Camila Borelli, 2014. "The bivariate Sinh-Elliptical distribution with applications to Birnbaum–Saunders distribution and associated regression and measurement error models," Computational Statistics & Data Analysis, Elsevier, vol. 80(C), pages 1-16.
    5. Carolina Marchant & Víctor Leiva & Francisco José A. Cysneiros & Juan F. Vivanco, 2016. "Diagnostics in multivariate generalized Birnbaum-Saunders regression models," Journal of Applied Statistics, Taylor & Francis Journals, vol. 43(15), pages 2829-2849, November.
    6. Naijun Sha, 2019. "A New Inference Approach for Type-II Generalized Birnbaum-Saunders Distribution," Stats, MDPI, vol. 2(1), pages 1-16, February.
    7. Vilca, Filidor & Romeiro, Renata G. & Balakrishnan, N., 2016. "A bivariate Birnbaum–Saunders regression model," Computational Statistics & Data Analysis, Elsevier, vol. 97(C), pages 169-183.
    8. Vilca, Filidor & Balakrishnan, N. & Zeller, Camila Borelli, 2014. "A robust extension of the bivariate Birnbaum–Saunders distribution and associated inference," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 418-435.
    9. Jochen Ranger & Jörg-Tobias Kuhn & José-Luis Gaviria, 2015. "A Race Model for Responses and Response Times in Tests," Psychometrika, Springer;The Psychometric Society, vol. 80(3), pages 791-810, September.

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