Generalized total time on test transform for weighted variables, properties and applications

Document Type : Special Issue Dedicated to memory of Prof. Mahbanoo Tata

Authors

1 Department of Statistics, Velayat University of Iranshahr, Iranshahr, Iran

2 Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

Abstract

In this article, the generalized  total  time on test transform and some related transforms for weighted variables are stated. Their characteristics and relationship with each other have been considered and also these transforms have been investigated in the weighted mode from the point of view of stochastic orders. Also, by presenting graphs of generalized total time on test transform   for some common weight functions, its behavior with respect to the weighted function is studied. Then the relationship of this transform with its initial state is expressed. In the following, the topic under discussion is explained with some practical examples. Then providing a comprehensive exploration of the applications of the studied transforms within the domains of insurance and reliability. By delving into these practical contexts, we gain valuable insights into how these mathematical tools can be effectively utilized to address complex challenges in risk assessment, decision-making, and resource allocation. Additionally, the examination of the NBU class of distributions offers a deeper understanding of their behavior, shedding light on their relevance and applicability in various statistical analyses.
    Finally, the article concludes with a detailed discussion of a specific real dataset, offering a concrete demonstration of how the topic under study can be applied in practice.

Keywords

Main Subjects

References

[1] Abdul Sathar, E. I., & Nair, R. D. (2023). A study on weighted dynamic survival and failure extropies. Communications in Statistics-Theory and Methods, 52(3), 623{642. https://doi.org/10.1080/03610926.2021.1919308
[2] Barlow, R. E, & Campo, R. (1975), Total Time on Test Processes and Applications to Failure Data Analysis, In:Reliability and Fault Tree Analysis.SIAM,Philadelphia,PA.
[3] Barlow, R. E, & Proschan, F. (1975). Statistical theory of reliability and life testing: probability models. Florida State Univ Tallahassee.
[4] Bergman, B., & Klefsjo, B. (1982). A graphical method applicable to agereplacement problems. IEEE Transactions on Reliability, 31(5), 478{481. https://doi.org/10.1109/TR.1982.5221439
[5] Bergman, B, & Klefsjo, B. (1984). The total time on test concept and its use in reliability theory. Operations Research, 32(3), 596{606. https://doi.org/10.1287/opre.32.3.596
[6] Chakraborty, S, & Pradhan, B. (2023). On weighted cumulative Tsallis residual and past entropy measures.  Communications in Statistics-Simulation and Computation, 52(5), 2058{2072. https://doi.org/10.1080/03610918.2021.1897623
[7] Esfahani, M, Amini, M, & Mohtashami, Borzadaran, G. R. (2021). On the applications of total time on test transform in reliability. Journal of Statistical Sciences, 15(1), 1{23. https://doi.org/10.52547/jss.15.1.1
[8] Esfahani, M., Mohtashami-Borzadaran, GR, & Amini, M. (2023). Theoretical aspects of total time on test transform of weighted variables and applications. Kybernetika, 59(2), 209-23 doi:10.14736/kyb-2023-2-02093.
[9] Fisher, R. A. (1934). The E ect of Methods of Ascertainment Upon the Estimation of Frequencies. Annals of Eugenics, 6, 13{25. https://doi.org/10.1111/j.1469-1809.1934.tb02105.x
[10] Gamiz, M. L., Nozal-Canadas, R, & Raya-Miranda, R. (2020). TTT-SiZer: A graphic tool for aging trends recognition. Reliability Engineering and System Safety, 202, 107010. https://doi.org/10.1016/j.ress.2020.107010
[11] Jewitt, I. (1989). Choosing between risky prospects: the characterization of comparative statics results, and location independent risk. Management Science, 35(1), 60{70. https://doi.org/10.1287/mnsc.35.1.60
[12] Kayid, M. (2007). A general family of NBU classes of life distributions. Statistical methodology, 4(2), 185{195. https://doi.org/10.1016/j.stamet.2006.06.002
[13] Klefsjo, B. (1982). On aging properties and total time on test transforms. Scandinavian Journal of Statistics, 37{41.
[14] Knopik, L. (2005). Some results on the ageing class. Control and Cybernetics, 34(4), 1175{1180.
[15] Kochar, S. C., Li, X, & Shaked, M. (2002), The total time on test transform and the excess wealth stochastic orders of distributions, Advances in Applied Probability, 34, 826{845. https://doi.org/10.1239/aap/1037990955
[16] Li, X., & Shaked, M. (2007), A General Family of Univariate Stochastic Orders, Journal of Statistical Planning and Inference, 137, 3601{3610. https://doi.org/10.1016/j.jspi.2007.03.035
[17] Marshall, A. W., & Proschan, F. (1964). An inequality for convex functions involving majorization (p. 0008). Mathematics Research Laboratory, Boeing Scienti c Research Laboratories.
[18] Nair, N. U., Sankaran, PG, & Balakrishnan, N. (2013), Quantile-Based Reliability Analysis, Basel: Birkhauser
[19] Nguyen, Thuan, & Nguyen, Thinh. (2020). A linear time partitioning algorithm for frequency weighted impurity functions. In ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (5375{5379). IEEE. https://doi.org/10.1109/ICASSP40776.2020.9054763
[20] Nichols, M. D., & Padgett, W. J. (2006). A bootstrap control chart for Weibull percentiles. Quality and reliability engineering international, 22(2), 141{151. https://doi.org/10.1002/qre.691
[21] Patil, G. P. (2014). Weighted distributions. Wiley StatsRef: Statistics Reference online.
[22] Pellerey, F., & Shaked, M. (1997). Characterizations of the IFR and DFR aging notions by means of the dispersive order. Statistics and Probability Letters, 33(4), 389{393. https://doi.org/10.1016/S0167-7152(96)00152-6
[23] Rao, C. R. (1965). On discrete distributions arising out of methods of ascertainment. Sankhya: The Indian Journal of Statistics, Series A, 311{324.
[24] Saghir, A., Hamedani, GG, Tazeem, S., & Khadim, A. (2017). Weighted distributions: A brief review, perspective and characterizations. International Journal of Statistics and Probability, 7(1), 39{71. https://doi.org/10.5539/ijsp.v6n3p109
[25] Shaked, M., & Shanthikumar, J. G. (2007), Stochastic Orders, Springer Science and Business Media.
[26] Shaked, M., Sordo, MA, & Suarez-Llorens, A. (2010). A class of location-independent variability orders, with applications. Journal of applied probability, 47(2), 407{425. https://doi.org/10.1239/jap/1276784900
[27] Sordo, M. A. (2009). On the relationship of location-independent riskier order to the usual stochastic order. Statistics and Probability Letters, 79(2), 155{157. https://doi.org/10.1016/j.spl.2008.07.033
[28] Yan, L., Kang, D, & Wang, H. (2021). Further results of the TTT transform ordering of order n. Symmetry, 13(10), 1960. https://doi.org/10.3390/sym13101960
Journal of Mahani Mathematical Research

Articles in Press, Accepted Manuscript
Available Online from 15 June 2024

Files

History

  • Receive Date: 17 November 2023
  • Revise Date: 13 April 2024
  • Accept Date: 14 June 2024

Share

How to cite

Statistics