Skip to main content
SpringerLink
Log in

Tsallis statistics in reliability analysis: Theory and methods

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract.

Tsallis statistics, which is based on a non-additive entropy characterized by an index q, is a very useful tool in physics and statistical mechanics. This paper presents an application of Tsallis statistics in reliability analysis. We first show that the q-gamma and incomplete q-gamma functions are q-generalized. Then, three commonly used statistical distributions in reliability analysis are introduced in Tsallis statistics, and the corresponding reliability characteristics including the reliability function, hazard function, cumulative hazard function and mean time to failure are investigated. In addition, we study the statistical inference based on censored reliability data. Specifically, we investigate the point and interval estimation of the model parameters of the q-exponential distribution based on the maximum likelihood method. Simulated and real-life datasets are used to illustrate the methodologies discussed in this paper. Finally, some concluding remarks are provided.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C. Tsallis, Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World (Springer, New York, 2009)

  2. T. Yamano, Physica A 305, 486 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  3. J.-F. Bercher, C. Vignat, Physica A 387, 5422 (2008)

    Article  ADS  Google Scholar 

  4. S. Martinez, F. Pennini, A. Plastino, Phys. Lett. A 278, 47 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  5. A.S. Parvan, Eur. Phys. J. A 51, 108 (2015)

    Article  ADS  Google Scholar 

  6. Z. Liu, Q. Hu, Y. Cui, Q. Zhang, Neurocomputing 142, 393 (2014)

    Article  Google Scholar 

  7. S. Furuichi, J. Math. Phys. 47, 023302 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  8. Z. Darócy, Inform. Control 16, 36 (1970)

    Article  Google Scholar 

  9. S. Amari, Information Geometry and Its Applications (Springer, Japan, 2016)

  10. S. Amari, A. Ohara, Entropy 13, 1170 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  11. A. Ohara, Phys. Lett. A 370, 184 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  12. H. Matsuzoe, Differ. Geom. Appl. 35, 323 (2014)

    Article  MathSciNet  Google Scholar 

  13. G. Loaiza, H.R. Quiceno, J. Math. Anal. Appl. 398, 466 (2013)

    Article  MathSciNet  Google Scholar 

  14. K.P. Nelson, S. Umarov, Physica A 389, 2157 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  15. M. Jauregui, C. Tsallis, Phys. Lett. A 375, 2085 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  16. J. Huang, W.A. Yong, L. Hong, J. Math. Anal. Appl. 436, 501 (2016)

    Article  MathSciNet  Google Scholar 

  17. N. Balakrishnan, R. Aggarwala, Progressive Censoring: Theory, Methods, and Applications (Birkhäuser, Boston, 2000)

  18. K.S. Trivedi, Probability and Statistics with Reliability, Queuing, and Computer Science Applications, 2nd edition (John Wiley & Sons, New Jersey, 2001)

  19. E.T. Lee, J.W. Wang, Statistical Methods for Survival Data Analysis 3rd edition (John Wiley & Sons, New Jersey, 2003)

  20. W. Nelson, Accelerated Testing: Statistical Models, Test Plans, and Data Analyses (John Wiley & Sons, New York, 1990)

  21. W.Q. Meeker, L.A. Escobar, Statistical Methods for Reliability Data (John Wiley & Sons, New York, 1998)

  22. V. Kumar, Physica A 462, 667 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  23. A.M. Mathai, S.B. Provost, IEEE Trans. Reliab. 55, 237 (2006)

    Article  Google Scholar 

  24. X.P. Zhang, J.Z. Shang, X. Chen, C.H. Zhang, Y.S. Wang, IEEE Trans. Reliab. 63, 764 (2014)

    Article  Google Scholar 

  25. D. Han, H.K.T. Ng, Commun. Stat. Theory Methods 43, 2384 (2014)

    Article  MathSciNet  Google Scholar 

  26. S. Nadarajah, S. Kotz, Physica A 377, 465 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  27. S. Picoli Jr., R.S. Mendes, L.C. Malacarne, Physica A 324, 678 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  28. J. Ludescher, C. Tsallis, A. Bunde, EPL 95, 68002 (2011)

    Article  ADS  Google Scholar 

  29. W. Weibull, J. Appl. Mech. 18, 293 (1951)

    Google Scholar 

  30. S.J. Almalki, S. Nadarajah, Reliab. Eng. Syst. Saf. 124, 32 (2014)

    Article  Google Scholar 

  31. C.D. Lai, D.N.P. Murthy, M. Xie, WIREs Comput. Stat. 3, 282 (2011)

    Article  Google Scholar 

  32. C.F. Zhang, Y.M. Shi, M. Wu, J. Comput. Appl. Math. 297, 65 (2016)

    Article  MathSciNet  Google Scholar 

  33. A.C. Cohen, Technometrics 5, 327 (1963)

    Article  MathSciNet  Google Scholar 

  34. N. Balakrishnan, E. Cramer, The Art of Progressive Censoring: Applications to Reliability and Quality (Birkhäuser, Basel, 2004)

  35. P.S. Chan, H.K.T. Ng, F. Su, Metrika 78, 747 (2015)

    Article  MathSciNet  Google Scholar 

  36. S. Park, H.K.T. Ng, P.S. Chan, Stat. Probab. Lett. 97, 142 (2015)

    Article  MathSciNet  Google Scholar 

  37. F. Zhang, Y. Shi, Physica A 446, 234 (2016)

    Article  MathSciNet  Google Scholar 

  38. F. Zhang, Y. Shi, SpringerPlus 5, 1 (2016)

    Article  Google Scholar 

  39. K. Soetaert, rootSolve: Nonlinear root finding, equilibrium and steady-state analysis of ordinary differential equations, R-package version 1.6 (2009)

  40. B. Efron, The Jackknife, the Bootstrap and Other Re-sampling plans, in CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 38 (SIAM, Philadelphia, PA, 1982)

  41. D. Kundu, A. Joarder, Comput. Stat. Data Anal. 50, 2509 (2006)

    Article  MathSciNet  Google Scholar 

  42. N. Balakrishnan, R.A. Sandhu, Am. Stat. 49, 229 (1995)

    Google Scholar 

  43. J. Buckley, I. James, Biometrika 66, 429 (1979)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Northwestern Polytechnical University, 710072, Shaanxi, China

    Fode Zhang, Yimin Shi & Ruibing Wang

  2. Department of Statistical Science, Southern Methodist University Dallas, 75275-0332, Dallas, TX, USA

    Fode Zhang & Hon Keung Tony Ng

Authors
  1. Fode Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yimin Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hon Keung Tony Ng
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ruibing Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fode Zhang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, F., Shi, Y., Keung Tony Ng, H. et al. Tsallis statistics in reliability analysis: Theory and methods. Eur. Phys. J. Plus 131, 379 (2016). https://doi.org/10.1140/epjp/i2016-16379-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/i2016-16379-8

Search

Navigation