The most common non-monotonic hazard rate situations in life sciences and engineering involves bathtub shapes. This paper focuses on the quantile residual life function in the class of lifetime distributions that have bathtub-shaped hazard rate functions. For this class of distributions, the shape of the $ \alpha $-quantile residual lifetime function was studied. Then, the change points of the $ \alpha $-quantile residual life function of a general weighted hazard rate model were compared with the corresponding change points of the basic model in terms of their location. As a special weighted model, the order statistics were considered and the change points related to the order statistics were compared with the change points of the baseline distribution. Moreover, some comparisons of the change points of two different order statistics were presented.
Citation: Mohamed Kayid. Some new results on bathtub-shaped hazard rate models[J]. Mathematical Biosciences and Engineering, 2022, 19(2): 1239-1250. doi: 10.3934/mbe.2022057
The most common non-monotonic hazard rate situations in life sciences and engineering involves bathtub shapes. This paper focuses on the quantile residual life function in the class of lifetime distributions that have bathtub-shaped hazard rate functions. For this class of distributions, the shape of the $ \alpha $-quantile residual lifetime function was studied. Then, the change points of the $ \alpha $-quantile residual life function of a general weighted hazard rate model were compared with the corresponding change points of the basic model in terms of their location. As a special weighted model, the order statistics were considered and the change points related to the order statistics were compared with the change points of the baseline distribution. Moreover, some comparisons of the change points of two different order statistics were presented.
[1] | S. Rajarshi, M. B. Rajarshi, Bathtub distributions: a review, Commun. Stat. Theory Methods, 17 (1988), 2597–2621. doi: 10.1080/03610928808829761. doi: 10.1080/03610928808829761 |
[2] | C. D. Lai, M. Xie, D. N. P. Murthy, Bathtub-shaped failure rate life distributions, Handb. Stat., 20 (2001), 69–104. doi: 10.1016/S0169-7161(01)20005-4. doi: 10.1016/S0169-7161(01)20005-4 |
[3] | R. E. Glaser, Bathtub and related failure rate characterizations, J. Am. Stat. Assoc., 75 (1980), 667–672. doi: 10.2307/2287666. doi: 10.2307/2287666 |
[4] | G. S. Mudholkar, D. K. Srivastava, Exponentiated Weibull family for analyzing bathtub failure-rate data, IEEE Trans. Reliab., 42 (1993), 299–302. doi: 10.1109/24.229504. doi: 10.1109/24.229504 |
[5] | J. Navarro, P. J. Hernandez, How to obtain bathtub-shaped failure rate models from normal mixtures, Probab. Eng. Inf. Sci., 18 (2004), 511–531. doi: 10.1017/S0269964804184076. doi: 10.1017/S0269964804184076 |
[6] | M. Xie, Y. Tang, T. N. Goh, A modified Weibull extension with bathtub-shaped failure rate function, Reliab. Eng. Syst. Saf., 76 (2002), 279–285. doi: 10.1016/S0951-8320(02)00022-4. doi: 10.1016/S0951-8320(02)00022-4 |
[7] | F. K. Wang, A new model with bathtub-shaped failure rate using an additive Burr XII distribution, Reliab. Eng. Syst. Saf., 70 (2000), 305–312. doi: 10.1016/S0951-8320(00)00066-1. doi: 10.1016/S0951-8320(00)00066-1 |
[8] | C. D. Lai, M. Xie, Stochastic Aging and Dependence for Reliability, Springer Science and Business Media, (2006). |
[9] | S. Nadarajah, Bathtub-shaped failure rate functions, Qual. Quant., 43 (2009), 855–863. doi: 10.1007/s11135-007-9152-9. doi: 10.1007/s11135-007-9152-9 |
[10] | J. Mi, Bathtub failure rate and upside-down bathtub mean residual life, IEEE Trans. Reliab., 44 (1995), 388–396. doi: 10.1109/24.406570. doi: 10.1109/24.406570 |
[11] | R. C. Gupta, H. O. Akman, Mean residual life functions for certain types of non-monotonic ageing, Commun. Stat. Stochastic Models, 11 (1995), 219–225. doi: 10.1080/15326349508807340. doi: 10.1080/15326349508807340 |
[12] | H. W. Block, T. H. Savits, H. Singh, A criteria for burn-in that balances mean residual life and residual variance, Oper. Res., 50 (2002), 290–296. doi: 10.1287/opre.50.2.290.435. doi: 10.1287/opre.50.2.290.435 |
[13] | A. L. Haines, N. D. Singpurwalla, Some contributions to the stochastic characterization of wear, in Reliability and Biometry, (1974), 47–80. |
[14] | B. C. Arnold, P. L. Brockett, When does the $ \beta $ th percentile residual life function determine the distribution?, Oper. Res., 31 (1983), 391–396. doi: 10.2307/170808. doi: 10.2307/170808 |
[15] | R. C. Gupta, E. S. Langford, On the determination of a distribution by its median residual life function: A functional equation, J. Appl. Probab., 21 (1984), 120–128. doi: 10.2307/3213670. doi: 10.2307/3213670 |
[16] | H. Joe, F. Proschan, Percentile residual life functions, Oper. Res., 32 (1984), 668–678. doi: 10.1287/opre.32.3.668. doi: 10.1287/opre.32.3.668 |
[17] | H. Joe, Characterizations of life distributions from percentile residual lifetimes, Ann. Inst. Stat. Math., 37 (1985), 165–172. doi: 10.1007/BF02481089. doi: 10.1007/BF02481089 |
[18] | R. L. Launer, Graphical techniques for analyzing failure data with the percentile residual-life function, IEEE Trans. Reliab., 42 (1993), 71–75. doi: 10.1109/24.210273. doi: 10.1109/24.210273 |
[19] | J. K. Song, G. Y. Cho, A note on percentile residual life, Sankhyā, 57 (1995), 333–335. |
[20] | G. D. Lin, On the characterization of life distributions via percentile residual lifetimes, Sankhyā, 71 (2009), 64–72. |
[21] | M. S. Noughabi, M. Kayid, Bivariate quantile residual life: a characterization theorem and statistical properties, Stat. Pap., 60 (2019), 2001–2012. doi: 10.1007/s00362-017-0905-9. doi: 10.1007/s00362-017-0905-9 |
[22] | M. Kayid, M. S. Noughabi, A. M. Abouammoh, A nonparametric estimator of bivariate quantile residual life model with application to tumor recurrence data set, J. Classif., 37 (2020), 237–253. doi: 10.1007/s00357-018-9300-z. doi: 10.1007/s00357-018-9300-z |
[23] | M. S. Noughabi, M. Kayid, A. M. Abouammoh, Dynamic multivariate quantile residual life in reliability theory, Math. Probl. Eng., 2018 (2018), 10. doi: 10.1155/2018/1245656. doi: 10.1155/2018/1245656 |
[24] | M. S. Noughabi, A. M. Franco-Pereira, Estimation of monotone bivariate quantile residual life, Stat., 52 (2018), 919–933. doi: 10.1080/02331888.2018.1470180. doi: 10.1080/02331888.2018.1470180 |
[25] | M. S. Noughabi, A. M. Franco-Pereira, Estimation of two ordered quantile residual life functions based on mixtures, J. Stat. Comput. Simul., 2021 (2021), 1–22. doi: 10.1080/00949655.2021.1947277. doi: 10.1080/00949655.2021.1947277 |
[26] | A. M. Franco-Pereira, R. E. Lillo, J. Romo, Characterization of bathtub distributions via percentile residual life functions, in Working paper, Universidad Carlos III de Madrid, (2010), 10–26. |
[27] | Y. Shen, M. Xie, L. C. Tang, On the change point of the mean residual life of series and parallel systems, Aust. N. Z. J. Stat., 52 (2010), 109–121. doi: 10.1111/j.1467-842X.2010.00569.x. doi: 10.1111/j.1467-842X.2010.00569.x |
[28] | M. S. Noughabi, G. Borzadaran, A. Roknabadi, On the reliability properties of some weighted models of bathtub shaped hazard rate distributions, Probab. Eng. Inf. Sci., 27 (2013), 125–140. doi: 10.1017/S0269964812000344. doi: 10.1017/S0269964812000344 |
[29] | K. Takahasi, A note on hazard rates of order statistics, Commun. Stat. Theory Methods, 17 (1988), 4133–4136, doi: 10.1080/03610928808829862. doi: 10.1080/03610928808829862 |
[30] | H. N. Nagaraja, Some reliability properties of order statistics, Commun. Stat. Theory Methods, 19 (1990), 307–316, doi: 10.1080/03610929008830202. doi: 10.1080/03610929008830202 |
[31] | N. Misra, M. Manoharan, H. Singh, Preservation of some ageing properties by order statistics, Probab. Eng. Inf. Sci., 7 (1993), 437–440. doi: 10.1017/S026996480000303X. doi: 10.1017/S026996480000303X |
[32] | J. Navarro, Y. Águilab, M. A. Sordoc, A. Suárez-Llorensc, Preservation of reliability classes under the formation of coherent systems, Appl. Stoch. Models Bus. Ind., 30 (2014), 444–454. doi: 10.1002/asmb.1985. doi: 10.1002/asmb.1985 |
[33] | F. J. Samaniego, On closure of the IFR class under formation of coherent systems, IEEE Trans. Reliab., 34 (1985), 69-72. doi: 10.1109/TR.1985.5221935. doi: 10.1109/TR.1985.5221935 |
[34] | M. Shaked, J. G. Shanthikumar, Stochastic Orders, Springer, (2007). |