Brazilian Journal of Probability and Statistics

Bivariate sinh-normal distribution and a related model

Debasis Kundu

Full-text: Open access

Abstract

Sinh-normal distribution is a symmetric distribution with three parameters. In this paper, we introduce bivariate sinh-normal distribution, which has seven parameters. Due to presence of seven parameters it is a very flexible distribution. Different properties of this new distribution has been established. The model can be obtained as a bivariate Gaussian copula also. Therefore, using the Gaussian copula property, several properties of this proposed distribution can be obtained. Maximum likelihood estimators cannot be obtained in closed forms. We propose to use two step estimators based on Copula, which can be obtained in a more convenient manner. One data analysis has been performed to see how the proposed model can be used in practice. Finally, we consider a bivariate model which can be obtained by transforming the sinh-normal distribution and it is a generalization of the bivariate Birnbaum–Saunders distribution. Several properties of the bivariate Birnbaum–Saunders distribution can be obtained as special cases of the proposed generalized bivariate Birnbaum–Saunders distribution.

Article information

Source
Braz. J. Probab. Stat., Volume 29, Number 3 (2015), 590-607.

Dates
Received: July 2013
Accepted: December 2013
First available in Project Euclid: 11 June 2015

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1433983067

Digital Object Identifier
doi:10.1214/13-BJPS235

Mathematical Reviews number (MathSciNet)
MR3355749

Zentralblatt MATH identifier
1326.62028

Keywords
Birnbaum–Saunders distribution bivariate Birnbaum–Saunders distribution log-Birnbaum–Saunders distribution maximum likelihood estimators copula two stage estimators total positivity of order two

Citation

Kundu, Debasis. Bivariate sinh-normal distribution and a related model. Braz. J. Probab. Stat. 29 (2015), no. 3, 590--607. doi:10.1214/13-BJPS235. https://projecteuclid.org/euclid.bjps/1433983067


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References

  • Birnbaum, Z. W. and Saunders, S. C. (1969). A new family of life distributions. Journal of Applied Probability 6, 319–327.
  • Diaz-Garcia, J. A. and Dominguez-Molina, J. R. (2006). Some generalizations of Birnbaum–Saunders and sinh-normal distributions. International Mathematical Forum 1, 1709–1727.
  • Gupta, P. L. and Gupta, R. C. (1997). On the multivariate normal hazard. Journal of Multivariate Analysis 62, 64–73.
  • Johnson, R. A. and Wichern, D. W. (1992). Applied Multivariate Statistical Analysis, 3rd ed. New Jersey, Englewood Cliffs: Prentice Hall.
  • Joe, H. (1997). Multivariate Model and Dependence Concepts. London: Chapman and Hall.
  • Joe, H. (2005). Asymptotic efficiency of the two-stage estimation method for copula-based models. Journal of Multivariate Analysis 94, 401–419.
  • Kundu, D., Balakrishnan, N. and Jamalizadeh, A. (2010). Bivariate Birnbaum–Saunders distribution and its associated inference. Journal of Multivariate Analysis 101, 113–125.
  • Marshall, A. W. (1975). Some comments on the hazard gradient. Stochastic Processes and Its Applications 3, 293–300.
  • Meyer, C. (2013). The bivariate normal copula. Communications in Statistics—Theory and Methods 42, 2402–2422.
  • Nelsen, R. B. (2006). An Introduction to Copulas. New York: Springer.
  • Owen, W. J. (2004). Another look at the Birnbaum–Saunders distribution. Available at http://www.stat.lanl.gov/MMR2004/Extended%20Abstract/WOwnn.pdf.
  • Rieck, J. R. (1989). Statistical analysis for the Birnbaum–Saunders fatigue life distribution. Ph.D. thesis, Clemson Univ., Dept. Mathematical Sciences, Canada.
  • Rieck, J. R. and Nedelman, J. R. (1991). A log-linear model for the Birnbaum–Saunders distribution. Technometrics 33, 51–60.