Abstract

In the literature, two-parameter distributions which exhibit all three types of decreasing, increasing and bathtub shape hazard rate functions are very few. In this paper, we propose a new two-parameter distribution, called Gompertz-weighted exponential distribution, having these three types of hazard rate functions. The proposed distribution is obtained by mixing the frailty parameter of the Gompertz distribution by weighted exponential distribution. The parameters are estimated by the maximum likelihood method and their performance is examined by extensive simulation studies. Three real data applications are provided to illustrate the superiority of the proposed distribution over many well known two-parameter distributions.

Citation details of the article

Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 33
Issue: 3
Year: 2020

DOI: 10.12732/ijam.v33i3.11

Download Section

Download the full text of article from here.

You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.

References

1. [1] S. Rajarshi, M. Rajarshi, Bathtub distributions: A review, Communications in Statistics-Theory and Methods, 17 (1988), 2597-2621.
2. [2] C. Lai, M. Xie, D.N.P. Murthy, Bathtub-shaped failure rate life distributions, In: Handbook of Statistics, 20, Elsevier, Amsterdam (2001), 69-104.
3. [3] S. Nadarajah, Bathtub-shaped failure rate functions, Quality and Quantity, 43 (2009), 855-863.
4. [4] I.W. Burr, Cumulative frequency functions, Annals of Mathematical Statistics , 13 (1942), 215-232.
5. [5] R.M. Smith, L.J. Bain, An exponential power life-testing distribution, Communications in Statistics-Theory and Methods, 4 (1975), 469-481.
6. [6] S. Paranjpe, M.B. Rajarshi, Modelling non-monotonic survivorship data with bathtub distributions, Ecology, 67 (1986), 1693-1695.
7. [7] Z. Chen, A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function, Statistics and Probability Letters, 49 (2000), 155-161.
8. [8] M.E. Ghitany, F. Alqallaf, D.K. Al-Mutairi, H.A. Husain, A two-parameter weighted Lindley distribution and its applications to survival data, Mathematics and Computers in Simulation, 81 (2011), 1190-1201.
9. [9] S. Nadarajah, H.S. Bakouch, R. Tahmasbi, A generalized Lindley distribution, Sankhya B, 73 (2011), 331-359.
10. [10] A. Marshall, I. Olkin, A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families, Biometrika, 84 (1997), 641-652.
11. [11] M. Bryson, M.M. Siddiqui, Some criteria for aging, Journal of the American Statistical Association, 64 (1969), 1472-1483.
12. [12] R.C. Gupta, H. Akman, Mean residual life function for certain types of non-monotonic ageing, Communications in Statistics-Stochastic Models, 11 (1995), 219-225.
13. [13] L. Lewin, Polylogarithms and associated functions, North Holland, New York (1981).
14. [14] B. Arnold, N. Balakrishnan, H. Nagaraja, A First Course in Order Statistics , Wiley, New York (1992).
15. [15] M. Shaked, J. Shanthikumar, Stochastic Orders, Springer, New York (2007).
16. [16] D. Cox, D. Hinkley, Theoretical Statistics, Chapman and Hall, New York (1974).
17. [17] R Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria (2018).
18. [18] P. Gupta, R.C. Gupta, R.D. Gupta, Modeling failure time data by Lehman alternatives, Communications in Statistics-Theory and Methods, 27 (1998), 887-904.
19. [19] F. Proschan, Theoretical explanation of observed decreasing failure rate, Technometrics, 5 (1963), 375-383.
20. [20] C. Escalante-Sandoval, Mixed distributions in low frequency analysis, Ingeniera, investigacin y tecnologa, 10 (2009), 247-253.