Research Article
BibTex RIS Cite
Year 2019, , 99 - 111, 30.08.2019
https://doi.org/10.33187/jmsm.439873

Abstract

References

  • [1] Al-khedhari A, Sarhan AM and Tadj L. Estimation of the generalized Rayleigh distribution parameters, International Journal of Reliability and Applications, (2008), 7(1), 1-12.
  • [2] Akaike H. Fitting autoregressive model for regression. Annals of the Institute of Statistical Mathematics, (1969), 21, 243-247.
  • [3] Bemis B, Bain LJ and Higgins JJ. Estimation and hypothesis testing for the parameters of a bivariate exponential distribution, Journal of the American Statistical Association, (1972), 67, 927-929.
  • [4] Hastings WK. Monte Carlo sampling methods using Markov chains and their applications. Biometrika, (1970), 57, 97-109.
  • [5] Kundu D, Sarhan AM and Gupta RD. On Sarhan-Balakrishnan bivariate distribution. Journal of Statistics and Probability, (2012), 1, no. 3, 163-170.
  • [6] Kundu D and Gupta RD. Bivariate Generalized Exponential Distribution, Journal of Multivariate Analysis, (2009), 100, 581-593.
  • [7] Lindley DV. Approximate Bayesian methods. Trabajos de Estadistica, (1980), 31, 223-245.
  • [8] Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E. Equation of state calculations by fast computing machines. Journal of Chemical Physics, (1953), 21, 1087-1092.
  • [9] Marshall AW and Olkin IA. A multivariate exponential distribution, Journal of American Statistical Association, (1967), 62, 30-44.
  • [10] Meintanis SG. Test of fit for Marshall-Olkin distributions with applications, Journal of Statistical Planning and inference, (2007), 137, 3954-3963.
  • [11] Mudholkar GS and Srivastava DK. Exponentiated Weibull family for analyzing bathtub failure-rate data, IEEE Transactions on Reliability, (1993), 42, 299-302.
  • [12] Raqab MZ, Kundu D. Burr Type X distribution; revisited, Journal of Probability and Statistical Science, (2006), 4(2), 179-193.
  • [13] Tierney L, Kadane JB. Accurate approximations for posterior moments and marginal densities. Journal of American Statistical Association, (1986), 81, 82-86.
  • [14] Sarhan AM and Balakrishnan N. A new class of bivariate distributions and its mixture. Journal of Multivariate Analysis, (2007), 98, 1508-1527.
  • [15] Surles JG and Padgett WJ. Inference for reliability and stress-strength for a scaled Burr Type X distribution, Lifetime Data Analysis, (2001), 7, 187-200.
  • [16] Surles JG and Padgett WJ. Some properties of a scaled Burr type X distribution, Journal of Statistical Planning and Inference, (2005), 128, 271-280.

The Bivariate Generalized Rayleigh Distribution

Year 2019, , 99 - 111, 30.08.2019
https://doi.org/10.33187/jmsm.439873

Abstract

This paper introduces a new bivariate distribution named the bivariate generalized Rayleigh distribution (BVGR). The proposed distribution is of type of Marshall-Olkin (MO) distribution. The BVGR distribution has  generalized Rayleigh marginal distributions. The joint cumulative distribution function,  the joint survival function, the joint probability density function and the joint hazard rate function of the proposed distribution are obtained in closed forms. Statistical properties of the BVGR distribution are investigated. The maximum likelihood and Bayes methods are applied to estimate the unknown parameters. Both maximum likelihood and Bayes estimates are not obtained analytically. Therefore, numerical algorithms are required to report on the model parameters and its reliability characteristics. Markov Chain Monte Carlo (MCMC) algorithm is applied for the Bayesian method. A real data set is analyzed using the proposed distribution and compared it with existing distributions. It is observed that the BVGR model fits this dataset better than the MO and the bivariate generalized exponential (BVGE) distributions.

References

  • [1] Al-khedhari A, Sarhan AM and Tadj L. Estimation of the generalized Rayleigh distribution parameters, International Journal of Reliability and Applications, (2008), 7(1), 1-12.
  • [2] Akaike H. Fitting autoregressive model for regression. Annals of the Institute of Statistical Mathematics, (1969), 21, 243-247.
  • [3] Bemis B, Bain LJ and Higgins JJ. Estimation and hypothesis testing for the parameters of a bivariate exponential distribution, Journal of the American Statistical Association, (1972), 67, 927-929.
  • [4] Hastings WK. Monte Carlo sampling methods using Markov chains and their applications. Biometrika, (1970), 57, 97-109.
  • [5] Kundu D, Sarhan AM and Gupta RD. On Sarhan-Balakrishnan bivariate distribution. Journal of Statistics and Probability, (2012), 1, no. 3, 163-170.
  • [6] Kundu D and Gupta RD. Bivariate Generalized Exponential Distribution, Journal of Multivariate Analysis, (2009), 100, 581-593.
  • [7] Lindley DV. Approximate Bayesian methods. Trabajos de Estadistica, (1980), 31, 223-245.
  • [8] Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E. Equation of state calculations by fast computing machines. Journal of Chemical Physics, (1953), 21, 1087-1092.
  • [9] Marshall AW and Olkin IA. A multivariate exponential distribution, Journal of American Statistical Association, (1967), 62, 30-44.
  • [10] Meintanis SG. Test of fit for Marshall-Olkin distributions with applications, Journal of Statistical Planning and inference, (2007), 137, 3954-3963.
  • [11] Mudholkar GS and Srivastava DK. Exponentiated Weibull family for analyzing bathtub failure-rate data, IEEE Transactions on Reliability, (1993), 42, 299-302.
  • [12] Raqab MZ, Kundu D. Burr Type X distribution; revisited, Journal of Probability and Statistical Science, (2006), 4(2), 179-193.
  • [13] Tierney L, Kadane JB. Accurate approximations for posterior moments and marginal densities. Journal of American Statistical Association, (1986), 81, 82-86.
  • [14] Sarhan AM and Balakrishnan N. A new class of bivariate distributions and its mixture. Journal of Multivariate Analysis, (2007), 98, 1508-1527.
  • [15] Surles JG and Padgett WJ. Inference for reliability and stress-strength for a scaled Burr Type X distribution, Lifetime Data Analysis, (2001), 7, 187-200.
  • [16] Surles JG and Padgett WJ. Some properties of a scaled Burr type X distribution, Journal of Statistical Planning and Inference, (2005), 128, 271-280.
There are 16 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Ammar Sarhan

Publication Date August 30, 2019
Submission Date July 2, 2018
Acceptance Date January 17, 2019
Published in Issue Year 2019

Cite

APA Sarhan, A. (2019). The Bivariate Generalized Rayleigh Distribution. Journal of Mathematical Sciences and Modelling, 2(2), 99-111. https://doi.org/10.33187/jmsm.439873
AMA Sarhan A. The Bivariate Generalized Rayleigh Distribution. Journal of Mathematical Sciences and Modelling. August 2019;2(2):99-111. doi:10.33187/jmsm.439873
Chicago Sarhan, Ammar. “The Bivariate Generalized Rayleigh Distribution”. Journal of Mathematical Sciences and Modelling 2, no. 2 (August 2019): 99-111. https://doi.org/10.33187/jmsm.439873.
EndNote Sarhan A (August 1, 2019) The Bivariate Generalized Rayleigh Distribution. Journal of Mathematical Sciences and Modelling 2 2 99–111.
IEEE A. Sarhan, “The Bivariate Generalized Rayleigh Distribution”, Journal of Mathematical Sciences and Modelling, vol. 2, no. 2, pp. 99–111, 2019, doi: 10.33187/jmsm.439873.
ISNAD Sarhan, Ammar. “The Bivariate Generalized Rayleigh Distribution”. Journal of Mathematical Sciences and Modelling 2/2 (August 2019), 99-111. https://doi.org/10.33187/jmsm.439873.
JAMA Sarhan A. The Bivariate Generalized Rayleigh Distribution. Journal of Mathematical Sciences and Modelling. 2019;2:99–111.
MLA Sarhan, Ammar. “The Bivariate Generalized Rayleigh Distribution”. Journal of Mathematical Sciences and Modelling, vol. 2, no. 2, 2019, pp. 99-111, doi:10.33187/jmsm.439873.
Vancouver Sarhan A. The Bivariate Generalized Rayleigh Distribution. Journal of Mathematical Sciences and Modelling. 2019;2(2):99-111.

29237    Journal of Mathematical Sciences and Modelling 29238

                   29233

Creative Commons License The published articles in JMSM are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.