Journal of the Iranian Statistical Society

Statistical Analysis of Bivariate Failure Time Data Based on Bathtub-shaped Failure Rate Model

Author

Shahid Beheshti University

10.29252/jirss.18.1.53

Abstract

Many distributions have been presented with bathtub-shaped failure rates for real-life data. A two-parameter distribution was defined by Chen (2000). This distribution can have a bathtub-shaped or increasing failure rate function. In this paper, we consider two bivariate models based on the proposed  distribution by Chen and use the proposed methods of Marshall and Olkin (1967) in the bivariate case and Marshall and Olkin (1997) in the univariate case. In the second case, their method is generalized to the bivariate case and a new bivariate distribution is introduced. These new bivariate distributions have natural interpretations, and they can be applied in fatal shock models or in competing risks models. We call these new distributions as the bivariate Chen (BCH) distribution and bivariate Chen-geometric (BCHG) distribution, respectively.  Moreover, the BCH can be obtained as a special case of the BCHG model. Then, the various properties of the new distributions are investigated.  The BCHG distribution has five parameters and the maximum likelihood estimators cannot be obtained in a closed form. We suggest using an EM algorithm that is very easy to implement. Also, Monte Carlo simulations are performed to investigate the effectiveness of the proposed algorithm. Finally, we analyze two real data sets for illustrative purposes.
 

Keywords

Aarset, MV. (1987), How to identify a bathtub hazard rate?. IEEE Transactions on Reliability, 36, 106 –108. Chen, Z . (2000), A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Statistics & Probability Letters, 49, 155–161. Dinse, G. (1982), Non-parametric estimation of partially incomplete time and types of failure data. Biometrics, 38, 417–431. Gaver, D.P and Acar, M. (1979), Analytical hazard representations for use in reliability, mortality, and simulation studies. Communications in Statistics, 8, 91–111. Gurvich, M.R and Dibenedetto, A.T and Rande, S.V.. (1997), A new statistical distribution for characterizing the random strength of brittle materials. Journal of Materials Science, 32, 2559–2564. Haynatzki, G.R and Weron, K and Haynatzka, V.R . (2000), A new statistical model of tumor latency time. Mathematical and Computer Modelling, 32, 251–256. Hjorth, U . (1980), A reliability distribution with increasing, decreasing, and bathtub-shaped failure rates. Technometrics, 22, 99–107. Kotz, S and Balakrishnan, N and Johnson, NL. (2000), Continuous multivariate distributions. New York: John Wiley and Sons. Kundu, D. (2004), Parameter estimation for partially complete time and type of failure data. Biometrical Journal, 46, 165–179. Kundu, D and Dey, A . (2009), Estimating the parameters of the Marshall-Olkin bivariate Weibull distribution by EM algorithm. Computational Statistics and Data Analysis, 35, 956–965. Kundu, D and Gupta, RD. (2006), Estimation of R = P(Y < X) for Weibull distribution. IEEE Transactions on Reliability, 55, 270–280. Leemis, L.M.(1986), Relationships among common univariate distributions. The American Statistician, 40, 143–146. Louis, TA. (1982), Finding the observed information matrix when using the EM algorithm. Journal of the Royal Statistical Society, 44, 226 –233. Marshall, A.W and Olkin, I . (1988), Families of multivariate distributions. Journal of the American Statistical Association, 83, 834–841. Marshall, A.W and Olkin, I . (1997), A new method of adding a parameter to a family of distributions with application to the exponential and Weibull families . Biometrika, 84, 641–652. Meintanis, SG. (2007), Test of fit for Marshall-Olkin distributions with applications. Journal of Statistical Planning and inference, 137, 3954–3963. Mudholkar, G.S and Srivastava, D.K . (1993), Exponentiated Weibull family for analyzing bathtub failure-ratedata. IEEE Transactions on Reliability, 42, 299–302. Nadaraja, S and Kotz, S . (2007), The two-parameter bathtub-shaped lifetime distribution. Quality and Reliability Engineering International, 23, 279–280. Smith, R.M and Bain, L.J. (1975),An exponential power life-testing distribution. Communications in Statistics, 4, 469–481. Wu, S.J . (2008), Estimation of the two-parameter bathtub-shaped lifetime distribution with progressive censoring. Journal of Applied Statistics, 35, 1139–1150. Wu, J.W. and Lu, H.L. and Chen, C.H and Wu, C.H . (2004), Statistical inference about the shape parameter of the new two-parameter bathtub-shaped lifetime distribution. Quality and Reliability Engineering International, 20, 607–616. Zhang, T and Xie, M Tang, L.C and Ng, S.H . (2005), Reliability and modeling of systems integrated with firmware and hard ware. International Journal of Reliability, Quality and Safety Engineering, 12, 227–239.
Journal of the Iranian Statistical Society
Volume 18, Issue 1
June 2019
Pages 53-87
  • Receive Date: 23 July 2022
  • Revise Date: 20 May 2024
  • Accept Date: 23 July 2022