| Title:
|
Theoretical aspects of total time on test transform of weighted variables and applications (English) |
| Author:
|
Esfahani, Mojtaba |
| Author:
|
Mohtashami-Borzadaran, Gholam Reza |
| Author:
|
Amini, Mohammad |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
59 |
| Issue:
|
2 |
| Year:
|
2023 |
| Pages:
|
209-233 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Although the total time on test (\textit{TTT}) transform is not a newly discovered concept, it has many applications in various fields. On the other hand, weighted distributions are extensively developed by the statisticians to tackle the insufficiency of the standard statistical distributions in modeling the arising data from real-world problems in the contexts like medicine, ecology, and reliability engineering. This paper develops the \textit{TTT} transform for the weighted random variables and investigates the behavior of the failure rate function of such variables based on the \textit{TTT} transform. In addition, the conditions for establishing the $TTT$ transform ordering for weight variables and its relationship with some stochastic orders have been investigated, and the conditions for establishing the \textit{TTT} transform order as well as the presentation of the new better than used in total time on test transform (\textit{NBUT}) class of the weighted variables have also been studied. Finally, by analyzing the real data sets, applications of the transform introduced in the fit of a model is presented, and it is shown that weighted models have a significant advantage over the base models. (English) |
| Keyword:
|
total time on test transform |
| Keyword:
|
generalized failure rate |
| Keyword:
|
generalized reversed failure rate |
| Keyword:
|
new better than used in total time on test transform |
| Keyword:
|
weight function |
| MSC:
|
62E15 |
| MSC:
|
62N05 |
| MSC:
|
62P99 |
| idMR:
|
MR4600375 |
| DOI:
|
10.14736/kyb-2023-2-0209 |
| . |
| Date available:
|
2023-06-19T09:03:50Z |
| Last updated:
|
2023-08-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151693 |
| . |
| Reference:
|
[1] Abd-Elfattah, A. M.: Goodness of fit test for the generalized Rayleigh distribution with unknown parameters..J. Statist. Comput. Simul. 81 (2011), 3, 357-366. MR 2772802, |
| Reference:
|
[2] Arnold, B. C.: Pareto Distributions Fairland..International Cooperative Publishing House. Maryland 1983. MR 0751409 |
| Reference:
|
[3] Barlow, R. E.: Geometry of the total time on test transform..Naval Res. Logist. Quarterly 26 (1979), 3, 393-402. MR 0546119, |
| Reference:
|
[4] Barlow, R. E., Campo, R.: Total time on test processes and applications to failure data analysis..In: Reliability and Fault Tree Analysis, SIAM, Philadelphia 1975. MR 0448771 |
| Reference:
|
[5] Bartoszewicz, J.: On a representation of weighted distributions..Statist. Probab. Lett. 79 (2009), 15, 1690-1694. MR 2547939, |
| Reference:
|
[6] Bartoszewicz, J., Skolimowska, M.: Preservation of classes of life distributions and stochastic orders under weighting..Statist. Probab. Lett. 76 (2006), 6, 587-596. MR 2255787, |
| Reference:
|
[7] Behdani, Z., Borzadaran, G. R. Mohtashami, Gildeh, B. S.: Relationship between the weighted distributions and some inequality measures..Commun. Statist. - Theory Methods 47 (2018), 22, 5573-5589. MR 3845363, |
| Reference:
|
[8] Behdani, Z., Borzadaran, G. R. Mohtashami, Gildeh, B. S.: Connection of generalized failure rate and generalized reversed failure rate with inequality curves..Int. J. Reliab. Quality Safety Engrg. 26 (2019), 02, 1950006. |
| Reference:
|
[9] Benduch-Fraszczak, M.: Some properties of the proportional odds model..Appl. Math. 37 (2010), 2, 247-256. MR 2650981, |
| Reference:
|
[10] Bergman, B., Klefsjö, B.: A graphical method applicable to age-replacement problems..IEEE Trans. Reliab. 31 (1982), 5, 478-481. |
| Reference:
|
[11] Bergman, B., Klefsjö, B.: The total time on test concept and its use in reliability theory..Oper. Res. 32 (1984), 3, 596-606. MR 0756006, |
| Reference:
|
[12] Bieniek, M.: Optimal bounds for the mean of the total time on test for distributions with decreasing generalized failure rate..Statistics 50 (2016), 6, 1206-1220. MR 3552989, |
| Reference:
|
[13] Bieniek, M., Szpak, M.: Sharp bounds for the mean of the total time on test for distributions with increasing generalized failure rate..Statistics 52 (2018), 4, 818-828. MR 3827420, |
| Reference:
|
[14] Cox, D. R.: Renewal Theory..Metrheun's Monograph. Barnes and Noble. Inc., New York 1962. MR 0153061 |
| Reference:
|
[15] Esfahani, M., Amini, M., Borzadaran, G. R. Mohtashami: On the applications of total time on test transform in reliability..J. Statist. Sci. 15 (2021), 1, 1-23. |
| Reference:
|
[16] Fagiuoli, E., Pellerey, F., Shaked, M.: A characterization of the dilation order and its applications..Statist. Papers 40 (1999), 4, 393-406. MR 1718261, |
| Reference:
|
[17] Franco-Pereira, A. M., Lillo, R. E., Shaked, M.: The decreasing percentile residual life ageing notion..Statistics 46 (2012), 5, 587-603. MR 2974640, |
| Reference:
|
[18] Franco-Pereira, A. M., Shaked, M.: The total time on test transform and the decreasing percentile residual life aging notion.Statist. Methodology 18 (2014), 32-40. MR 3151862, |
| Reference:
|
[19] Fisher, R. A.: The effect of methods of ascertainment upon the estimation of frequencies..Ann. Eugenics 6 (1934), 1), 13-25. |
| Reference:
|
[20] Gamiz, M. L., Nozal-Canadas, R., Raya-Miranda, R.: TTT-SiZer: A graphic tool for aging trends recognition..Reliab. Engrg. System Safety 202 (2020), 107010. |
| Reference:
|
[21] Haines, A. L., Singpurwalla, N. D.: Some contributions to the stochastic characterization of wear..Reliability and Biometry, Statistical Analysis of Lifelength (F. Proschan and R. J. Serfling, eds.), SIAM, Philadelphia 1974, pp. 47-80. MR 0353603 |
| Reference:
|
[22] Izadkhah, S., Rezaei, A. H., Amini, M., Borzadaran, G. R. Mohtashami: A general approach for preservation of some aging classes under weighting..Commun. Statistics-Theory Methods 42 (2013), 10, 1899-1909. MR 3045360, |
| Reference:
|
[23] Izadkhah, S., Amini, M., Borzadaran, G. R. Mohtashami: On preservation under univariate weighted distributions..Appl. Math. 60 (2015), 4, 453-467. MR 3396474, |
| Reference:
|
[24] Jewitt, I.: Choosing between risky prospects: the characterization of comparative statics results, and location independent risk..Management Sci. 35 (1989), 1, 60-70. MR 0985467, |
| Reference:
|
[25] Klefsjo, B.: On aging properties and total time on test transforms..Scandinav. J. Statist, (1982), 37-41. MR 0651859, |
| Reference:
|
[26] Klefsjo, B.: TTT-transforms - A useful tool when analysing different reliability problems..Reliab. Engrg. 15 (1986), 3, 231-241. |
| Reference:
|
[27] Klefsjo, B.: TTT-plotting - A tool for both theoretical and practical problems..J. Statist. Planning Inference 29 (1991), 1-2, 99-110. MR 1133694, |
| Reference:
|
[28] Klefsjo, B., Westberg, U.: TTT plotting and maintenance policies..Quality Engrg. 9 (1996), 2, 229-235. |
| Reference:
|
[29] Kochar, S. C., Li, X., Shaked, M.: The total time on test transform and the excess wealth stochastic orders of distributions..Adv. Appl. Probab. 34 (2002), 4, 826-845. MR 1938944, |
| Reference:
|
[30] Kumar, D., Westberg, U.: Maintenance scheduling under age replacement policy using proportional hazards model and ttt-plotting..Europ. J. Oper. Res. 99 (1997), 3, 507-515. |
| Reference:
|
[31] Lai, C. D., Xie, M.: Stochastic Ageing and Dependence for Reliability..Springer Science and Business Media, 2006. MR 2223811 |
| Reference:
|
[32] Marshall, A. W., Olkin, I.: A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families..Biometrika 84 (1997), 3, 641-652. Zbl 0888.62012, MR 1603936, |
| Reference:
|
[33] Misra, N., Gupta, N., Dhariyal, I. D.: Preservation of some aging properties and stochastic orders by weighted distributions..Commun. Statist. - Theory Methods 37 (2008), 5, 627-644. MR 2392347, |
| Reference:
|
[34] Nair, N. U., Sankaran, P. G.: Some new applications of the total time on test transforms..Statist. Methodology 10 (2013), 1, 93-102. MR 2974813, |
| Reference:
|
[35] Nair, N. U., Sankaran, P. G., Kumar, B. V.: Total time on test transforms of order n and their implications in reliability analysis..J. Appl. Probab. 45 (2008), 4, 1126-1139. MR 2484166, |
| Reference:
|
[36] Nguyen, T., Nguyen, T.: A linear time partitioning algorithm for frequency weighted impurity functions..In: ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), IEEE 2020, pp. 5375-5379. |
| Reference:
|
[37] Nichols, M. D., Padgett, W. J.: A bootstrap control chart for Weibull percentiles..Quality Reliab. Engrg. Int. 22 (2006), 2, 141-151. |
| Reference:
|
[38] Padgett, W. J., Spurrier, J. D.: Shewhart-type charts for percentiles of strength distributions..J. Quality Technol. 22 (1990), 283-288. |
| Reference:
|
[39] Patil, G. P.: Weighted Distributions..Wiley StatsRef: Statistics Reference Online, 2014. |
| Reference:
|
[40] Patil, G. P., Ord, J. K.: On size-biased sampling and related form-invariant weighted distributions..Sankhya 38B (1976), 48-61. MR 0652546, |
| Reference:
|
[41] Pham, T. G., Turkkan, N.: The Lorenz and the scaled total-time-on-test transform curves: a unified approach..IEEE Trans. Reliab. 43 (1994), 1, 76-84. |
| Reference:
|
[42] Rao, C. R.: On discrete distributions arising out of methods of ascertainment..Sankhya: Indian J. Statist. Ser. A, (1965), 311-324. MR 0208736 |
| Reference:
|
[43] Rao, K. R., Prasad, P. V. N.: Graphical methods for reliability of repairable equipment and maintenance planning..In: Proc. Annual Reliability and Maintainability Symposium, International Symposium on Product Quality and Integrity (Cat. No. 01CH37179), IEEE 2001, pp. 123-128. |
| Reference:
|
[44] Saghir, A., Hamedani, G. G., Tazeem, S., Khadim, A.: Weighted distributions: A brief review, perspective and characterizations..Int. J. Statist. Probab. 7 (2017), 1, 109-131. |
| Reference:
|
[45] Shaked, M., Shanthikumar, J. G.: Stochastic Orders..Springer Science and Business Media, 2007. MR 2265633 |
| Reference:
|
[46] Smith, R. L., Naylor, J. C.: A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution..Appl. Statist. 36 (1987), 358-369. MR 0918854, |
| Reference:
|
[47] Spiroiu, M. A.: Estimation of total time on test for large samples..In: MATEC Web of Conferences, EDP Sciences 2017, Vol. 112, p. 09008). |
| Reference:
|
[48] Sun, F. B., Kececloglu, D. B.: A new method for obtaining the TTT plot for a censored sample. In Annual Reliability and Maintainability..In: Proc. Symposium. (Cat. No. 99CH36283), IEEE 1999, pp. 112-117. |
| . |
