Gompertz-Exponential Distribution: Record Value Theory and Applications in Reliability

Abstract

The continuous probability distributions have wide applications in the field of transportation and reliability engineering. The continuous distributions are used to estimate how funds can be allocated to improve roads, railways, bridges, waterways, airports etc. and used to check the reliability/performance of a product. The Gompertz exponential (GoE) distribution is derived using Gompertz G generator. Some basic properties of the model have been derived. The parameters of the GoE distribution are estimated by maximum likelihood estimation method. The upper record values from the GoE distribution have also been introduced with various properties. Moreover, applications of the GoE distributions has been provided in the field of reliability to check the performance of some transportation related parts and the suggested model provides better than the existing well-known models. Finally, a simulation study is carried out. Random numbers of size 50 are generated 15 times for GoE distribution and upper records has been noted.

Keywords

• Gompertz family of distributions
• exponential
• MLE
• GoE
• reliability

References

1. Abdal-Hameed, M. Kh., Khaleel, M.A., Abdullah, Z.M., Oguntunde, P.E., Adejumo, A.O. (2018). Parameter estimation and reliability, hazard functions of Gompertz Burr Type XII distribution. Tikrit Journal for Administration and Economics Sciences, 1, 381-400.
2. Alexander, C., Cordeiro, G.M., Ortega, E.M.M. and Sarabia, J.M. (2012). Generalized beta- generated distributions. Computational Statistics and Data Analysis, 56, 1880-1897.
3. Alizadeh, M., Cordeiro, G.M., Bastos Pinho, L.G. and Ghosh, I. (2017). The Gompertz-g family of distributions. Journal of Statistical Theory and Practice, 11(1), 179{207. doi:10.1080/15598608.2016.1267668.
4. Alzaatreh, A., Lee, C. and Famoye, F. (2013). A new method for generating families of continuous distributions. Metron, 71, 63-79.
5. Alzaghal, A., Famoye, F. and Lee, C. (2013). Exponentiated T-X family of distributions with some applications. International Journal of Statistics and Probability, 2, 1-31.
6. Amini, M., MirMostafaee, S.M.T.K. and Ahmadi, J. (2012). Log-gamma-generated families of distributions. Statistics, 1, 1-20.
7. Bourguignon, M., Silva, R.B. and Cordeiro, G.M. (2014). The Weibull-G family of probability distributions. Journal of Data Science, 12, 53-68.
8. Chandler, K.N. (1952). The distribution and frequency of record values. Journal of Royal Statistical Society, B, 14, 220-228.

9. Cordeiro, G.M., Alizadeh, M. and Ortega, E.M. (2014). The exponentiated half-logistic family of distributions: Properties and applications. Journal of Probability and Statistics, 2014. doi:10.1155/2014/864396.
10. Cordeiro, G.M. and de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation, 81, 883-898.
11. Cordeiro, G.M., Ortega, E.M.M. and Silva, G.O. (2011). The exponentiated generalized gamma distribution with application to lifetime data. Journal of Statistical Computation and Simulation, 81, 827-842.
12. Eugene, N., Lee, C. and Famoye, F. (2002). Beta-normal distribution and its applications. Communications in Statistics-Theory and Methods, 31, 497-512.
13. Jones, M.C. (2004). Families of distributions arising from distributions of order statistics. Test, 13, 1-43.
14. Marshall, A.W. and Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika, 84, 641-652.
15. Morad A., Gauss M.C., Luis G.B.P. and Indranil, G. (2016). The Gompertz-G family of distributions. Journal of Statistical Theory and Practice, 11(1), 179-207.
16. Murthy, D.N.P., Xie, M. and Jiang, R. (2004). Weibull Models. Wiley.
17. Oguntunde, P.E., Adejumo, A.O., and Owoloko, E.A. (2017). Application of Kumaraswamy inverse exponential distribution to real lifetime data. International Journal of Applied Mathematics and Statistics, 56(5), 34-47.
18. Ramos, M.W., Marinho, P.R.D., Silva, R.V. and Cordeiro, G.M. (2013). The exponentiated Lomax Poisson distribution with an application to lifetime data. Advances and Applications in Statistics, 34, 107-135, 2013.
19. Risti c, M.M. and Balakrishnan, N. (2012). The gamma-exponentiated exponential distribution. Journal of Statistical Computation and Simulation, 82, 1191-1206.
20. Smith, R.I. and Naylor, J.C. (1987). A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution. Applied Statistics, 36, 258-369. doi:10.2307/2347795.
21. Torabi, H. and Hedesh, N.M. (2012). The gamma-uniform distribution and its applications. Kybernetik a, 1, 16-30.
22. Zografos, K. and Balakrishnan, N. (2009). On families of beta- and generalized gamma- generated distributions and associated inference. Statistical Methodology, 6, 344-362.