# Classical and Bayesian studies for a new lifetime model in presence of type-II censoring

• Goyal, Teena (Department of Mathematics & Statistics, Banasthali Vidyapith) ;
• Rai, Piyush K (Department of Statistics, Banaras Hindu University) ;
• Maury, Sandeep K (Department of Mathematics & Statistics, Banasthali Vidyapith)
• Received : 2019.02.08
• Accepted : 2019.05.08
• Published : 2019.07.31

#### Abstract

This paper proposes a new class of distribution using the concept of exponentiated of distribution function that provides a more flexible model to the baseline model. It also proposes a new lifetime distribution with different types of hazard rates such as decreasing, increasing and bathtub. After studying some basic statistical properties and parameter estimation procedure in case of complete sample observation, we have studied point and interval estimation procedures in presence of type-II censored samples under a classical as well as Bayesian paradigm. In the Bayesian paradigm, we considered a Gibbs sampler under Metropolis-Hasting for estimation under two different loss functions. After simulation studies, three different real datasets having various nature are considered for showing the suitability of the proposed model.

#### References

1. Aarset MV (1987). How to identify a bathtub hazard rate, IEEE Transactions on Reliability, 36, 106-108.
2. Aryal GR and Tsokos CP (2009). On the transmuted extreme value distribution with application, Nonlinear Analysis: Theory, Methods & Applications, 71, e1401-e1407. https://doi.org/10.1016/j.na.2009.01.168
3. Balakrishnan N and Cohen AC (2014). Order Statistics and Inference: Estimation Methods, Elsevier.
4. Box GEP and Tiao GC (1973). Bayesian Inference in Statistical Analysis, Addison-Wesley, Massachusetts.
5. Carpenter J and Bithell J (2000). Bootstrap confidence intervals: when, which, what? A practical guide for medical statisticians, Statistics in Medicine, 19, 1141-1164. https://doi.org/10.1002/(SICI)1097-0258(20000515)19:9<1141::AID-SIM479>3.0.CO;2-F
6. Chen MH and Shao QM (1999). Monte Carlo estimation of Bayesian credible and HPD intervals, Journal of Computational and Graphical Statistics, 8, 69-92.
7. Chen Z (2000). A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function, Statistics & Probability Letters, 49, 155-161. https://doi.org/10.1016/S0167-7152(00)00044-4
8. Chib S and Greenberg E (1995). Understanding the metropolis-hastings algorithm, The American Statistician, 49, 327-335.
9. Choulakian V and Stephens MA (2001). Goodness-of-fit tests for the generalized Pareto distribution, Technometrics, 43, 478-484. https://doi.org/10.1198/00401700152672573
10. Cohen AC (1965). Maximum likelihood estimation in theWeibull distribution based on complete and on censored samples, Technometrics, 7, 579-588. https://doi.org/10.1080/00401706.1965.10490300
11. Cordeiro GM, Ortega EMM, da Cunha DCC (2013). The exponentiated generalized class of distributions, Journal of Data Science, 11, 1-27.
12. Davison AC and Hinkley DV (1997). Bootstrap Methods and Their Application, Cambridge university press, Cambridge.
13. Dey S, Nassar M, and Kumar D (2017). ${\alpha}$ logarithmic transformed family of distributions with application, Annals of Data Science, 4, 457-482. https://doi.org/10.1007/s40745-017-0115-2
14. Dey S, Nassar M, Kumar D, and Alaboud F (2019). Alpha logarithmic transformed Frechet distribution: properties and estimation, Austrian Journal of Statistics, 48, 70-93.
15. Edwards W, Lindman H, and Savage LJ (1963). Bayesian statistical inference for psychological research., Psychological Review, 70, 193. https://doi.org/10.1037/h0044139
16. Efron B (1979). Bootstrap methods: another look at the jackknife, The Annals of Statistics, 7, 1-26. https://doi.org/10.1214/aos/1176344552
17. Efron B (1982). The Jackknife, the Bootstrap, and other Resampling Plans, 38, Siam.
18. Efron B and Tibshirani RJ (1994). An Introduction to the Bootstrap, CRC press, Florida.
19. Evans IG and Ragab AS (1983). Bayesian inferences given a type-2 censored sample from a Burr distribution, Communications in Statistics-Theory and Methods, 12, 1569-1580. https://doi.org/10.1080/03610928308828551
20. Gelfand AE and Smith AFM (1990). Sampling-based approaches to calculating marginal densities, Journal of the American Statistical Association, 85, 398-409. https://doi.org/10.1080/01621459.1990.10476213
21. Geman S and Geman D (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721-741.
22. Glaser RE (1980). Bathtub and related failure rate characterizations, Journal of the American Statistical Association, 75, 667-672. https://doi.org/10.1080/01621459.1980.10477530
23. Graham RL, Knuth DE, and Patashnik O (1994). Concrete Mathematics: A Foundation for Computer Science (2nd ed), Addison-Wesley, Mass.
24. Gupta RC, Gupta PL, and Gupta RD (1998). Modeling failure time data by Lehman alternatives, Communications in Statistics-Theory and Methods, 27, 887-904. https://doi.org/10.1080/03610929808832134
25. Gupta RD and Kundu D (2001). Exponentiated exponential family: an alternative to gamma and Weibull distributions, Biometrical Journal: Journal of Mathematical Methods in Biosciences, 43, 117-130.
26. Hastings WK (1970). Monte Carlo sampling methods using Markov chains and their applications, Biometrika, 57, 97-109. https://doi.org/10.1093/biomet/57.1.97
27. Hjorth U (1980). A reliability distribution with increasing, decreasing, constant and Bathtub-Shaped failure rates, Technometrics, 22, 99-107. https://doi.org/10.2307/1268388
28. Kumaraswamy P (1980). A generalized probability density function for double-bounded random processes, Journal of Hydrology, 46, 79-88. https://doi.org/10.1016/0022-1694(80)90036-0
29. Kundu D and Howlader H (2010). Bayesian inference and prediction of the inverse Weibull distribution for Type-II censored data, Computational Statistics & Data Analysis, 54, 1547-1558. https://doi.org/10.1016/j.csda.2010.01.003
30. Lawless JF (2011). Statistical Models and Methods for Lifetime Data, JohnWiley & Sons, New York.
31. Leiva V, Athayde E, Azevedo C, and Marchant C (2011). Modeling wind energy flux by a Birnbaum-Saunders distribution with an unknown shift parameter. Journal of Applied Statistics, 38, 2819-2838. https://doi.org/10.1080/02664763.2011.570319
32. Lindley DV (1958). Fiducial distributions and Bayes' theorem, Journal of the Royal Statistical Society. Series B (Methodological), 20, 102-107. https://doi.org/10.1111/j.2517-6161.1958.tb00278.x
33. Maurya SK, Kaushik A, Singh RK, Singh SK, and Singh U (2016). A new method of proposing distribution and its application to real data, Imperial Journal of Interdisciplinary Research, 2, 1331-1338.
34. Maurya SK, Kaushik A, Singh SK, and Singh U (2017). A new class of distribution having decreasing, increasing and bathtub-shaped failure rate, Communications in Statistics-Theory and Methods, 46, 10359-10372. https://doi.org/10.1080/03610926.2016.1235196
35. Maurya SK, Kumar D, Singh SK, and Singh U (2018). One parameter decreasing failure rate distribution, International Journal of Statistics & Economics, 19, 120-138.
36. Merovci F, Elbatal I, and Ahmed A (2013). Transmuted Generalized Inverse Weibull Distribution, arXiv preprint arXiv:1309.3268.
37. Merovci F and Puka L (2014). Transmuted Pareto distribution, ProbStat Forum, 7, 1-11.
38. Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, and Teller E (1953). Equation of state calculations by fast computing machines, The Journal of Chemical Physics, 21, 1087-1092. https://doi.org/10.1063/1.1699114
39. Metropolis N and Ulam S (1949). The Monte Carlo method, Journal of the American Statistical Association, 44, 335-341. https://doi.org/10.1080/01621459.1949.10483310
40. Mudholkar GS and Srivastava DK (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data, IEEE Transactions on Reliability, 42, 299-302. https://doi.org/10.1109/24.229504
41. Murthy DNP, Xie M, and Jiang R (2004). Weibull Models, John Wiley & Sons, Hoboken.
42. Nadarajah S, Bakouch HS, and Tahmasbi R (2011). A generalized Lindley distribution, Sankhya B, 73, 331-359. https://doi.org/10.1007/s13571-011-0025-9
43. Nadarajah S and Kotz S (2006). The exponentiated type distributions, Acta Applicandae Mathematica, 92, 97-111. https://doi.org/10.1007/s10440-006-9055-0
44. Nassar M, Afify AZ, Dey S, and Kumar D (2018). A new estension ofWeibull distribution: Properties and different methods of estimation, Journal of Computational and Applied Mathematics, 335, 1-18.
45. Nelson WB (2003). Recurrent Events Data Analysis for Product Repairs, Disease Recurrences, and other Applications (Volume 10), SIAM, London.
46. Pappas V, Adamidis K, and Loukas S (2012). A family of lifetime distributions, International Journal of Quality, Statistics, and Reliability, 2012, 1-6.
47. Peter H (1988). Theoretical comparision of Bootstrap confidence intervals, The Annals of Statistics, 16, 927-953. https://doi.org/10.1214/aos/1176350933
48. Robert C and Casella G (2013). Monte Carlo Statistical Methods, Springer Science & Business Media.
49. Shannon CE (1951). Prediction and entropy of printed English, Bell System Technical Journal, 30, 50-64. https://doi.org/10.1002/j.1538-7305.1951.tb01366.x
50. Shaw WT and Buckley IRC (2007). The Alchemy of Probability Distribution: Beyond Gram-Charlier Cornish-Fisher Expansions, and Skew-Normal and Kurtotic-Normal Distribution (Research report).
51. Sinha SK (1987). Bayesian estimation of the parameters and reliability function of a mixture of Weibull life distributions, Journal of Statistical Planning and Inference, 16, 377-387. https://doi.org/10.1016/0378-3758(87)90090-5
52. Singh SK, Singh U, and Kumar M (2013). Estimation of Parameters of Exponentiated Pareto Model for Progressive Type-II Censored Data with Binomial Removals Using Markov Chain Monte Carlo Method, International Journal of Mathematics & Computation, 21, 88-102.
53. Singh SK, Singh U, and Kumar M (2016). Bayesian estimation for Poisson-exponential model under progressive type-ii censoring data with binomial removal and its application to ovarian cancer data, Communications in Statistics-Simulation and Computation, 45, 3457-3475. https://doi.org/10.1080/03610918.2014.948189
54. Singh U, Gupta PK, and Upadhyay SK (2005). Estimation of parameters for exponentiated-Weibull family under type-II censoring scheme, Computational Statistics & Data Analysis, 48, 509-523. https://doi.org/10.1016/j.csda.2004.02.009
55. Smith AFM and Roberts GO (1993). Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods, Journal of the Royal Statistical Society. Series B (Methodological), 55, 3-23. https://doi.org/10.1111/j.2517-6161.1993.tb01466.x
56. Tierney L (1994). Markov chains for exploring posterior distributions, The Annals of Statistics, 22, 1701-1728. https://doi.org/10.1214/aos/1176325750
57. Varian HR (1975). A Bayesian approach to real estate assessment, Studies in Bayesian Econometric and Statistics in honor of Leonard J. Savage, 195-208, North Holland.