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Bivariate Birnbaum-Saunders distribution and associated inference

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

Abstract

Univariate Birnbaum-Saunders distribution has been used quite effectively to model positively skewed data, especially lifetime data and crack growth data. In this paper, we introduce bivariate Birnbaum-Saunders distribution which is an absolutely continuous distribution whose marginals are univariate Birnbaum-Saunders distributions. Different properties of this bivariate Birnbaum-Saunders distribution are then discussed. This new family has five unknown parameters and it is shown that the maximum likelihood estimators can be obtained by solving two non-linear equations. We also propose simple modified moment estimators for the unknown parameters which are explicit and can therefore be used effectively as an initial guess for the computation of the maximum likelihood estimators. We then present the asymptotic distributions of the maximum likelihood estimators and use them to construct confidence intervals for the parameters. We also discuss likelihood ratio tests for some hypotheses of interest. Monte Carlo simulations are then carried out to examine the performance of the proposed estimators. Finally, a numerical data analysis is performed in order to illustrate all the methods of inference discussed here.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:jmvana:v:101:y:2010:i:1:p:113-125
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    References listed on IDEAS

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    1. Ng, H.K.T. & Kundu, D. & Balakrishnan, N., 2006. "Point and interval estimation for the two-parameter Birnbaum-Saunders distribution based on Type-II censored samples," Computational Statistics & Data Analysis, Elsevier, vol. 50(11), pages 3222-3242, July.
    2. 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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    1. Zhu, Xiaojun & Balakrishnan, N., 2015. "Birnbaum–Saunders distribution based on Laplace kernel and some properties and inferential issues," Statistics & Probability Letters, Elsevier, vol. 101(C), pages 1-10.
    2. Debasis Kundu, 2017. "On Multivariate Log Birnbaum-Saunders Distribution," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 79(2), pages 292-315, November.
    3. 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.
    4. Kakizawa, Yoshihide, 2022. "Multivariate elliptical-based Birnbaum–Saunders kernel density estimation for nonnegative data," Journal of Multivariate Analysis, Elsevier, vol. 187(C).
    5. Palayangoda, Lochana K. & Ng, Hon Keung Tony, 2021. "Semiparametric and nonparametric evaluation of first-passage distribution of bivariate degradation processes," Reliability Engineering and System Safety, Elsevier, vol. 205(C).
    6. 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.
    7. Pan, Zhengqiang & Balakrishnan, Narayanaswamy, 2011. "Reliability modeling of degradation of products with multiple performance characteristics based on gamma processes," Reliability Engineering and System Safety, Elsevier, vol. 96(8), pages 949-957.
    8. Caro-Lopera, Francisco J. & Leiva, Víctor & Balakrishnan, N., 2012. "Connection between the Hadamard and matrix products with an application to matrix-variate Birnbaum-Saunders distributions," Journal of Multivariate Analysis, Elsevier, vol. 104(1), pages 126-139, February.
    9. 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.
    10. Rodrigo Puentes & Carolina Marchant & Víctor Leiva & Jorge I. Figueroa-Zúñiga & Fabrizio Ruggeri, 2021. "Predicting PM2.5 and PM10 Levels during Critical Episodes Management in Santiago, Chile, with a Bivariate Birnbaum-Saunders Log-Linear Model," Mathematics, MDPI, vol. 9(6), pages 1-24, March.
    11. Vilca, Filidor & Balakrishnan, N. & Zeller, Camila Borelli, 2014. "Multivariate Skew-Normal Generalized Hyperbolic distribution and its properties," Journal of Multivariate Analysis, Elsevier, vol. 128(C), pages 73-85.
    12. 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.
    13. 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.
    14. Guillermo Martínez-Flórez & Artur J. Lemonte & Germán Moreno-Arenas & Roger Tovar-Falón, 2022. "The Bivariate Unit-Sinh-Normal Distribution and Its Related Regression Model," Mathematics, MDPI, vol. 10(17), pages 1-26, August.
    15. Mahdi Teimouri, 2023. "Fast Bayesian Inference for Birnbaum-Saunders Distribution," Computational Statistics, Springer, vol. 38(2), pages 569-601, June.
    16. 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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