Skip to main content

On a Time Dependent Divergence Measure between Two Residual Lifetime Distributions

Mathematical Methods of Statistics volume 29, pages 135–148 (2020)Cite this article

Abstract

Recently, a time-dependent measure of divergence has been introduced by Mansourvar and Asadi (2020) to assess the discrepancy between the survival functions of two residual lifetime random variables. In this paper, we derive various time-dependent results on the proposed divergence measure in connection to other well-known measures in reliability engineering. The proposed criterion is also examined in mixture models and a general class of survival transformation models which results in some well-known models in the lifetime studies and survival analysis. In addition, the time-dependent measure is employed to evaluate the divergence between the lifetime distributions of \(k\)-out-of-\(n\) systems and also to assess the discrepancy between the distribution functions of the epoch times of a non-homogeneous Poisson process.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

Fig. 1
Fig. 2

REFERENCES

  1. S. M. Ali and S. D. Silvey, ‘‘A general class of coefficients of divergence of one distribution from another,’’ J. R. Stat. Soc. Series B Stat. Methodol. 28 (1), 131–142 (1996).

    MathSciNet  MATH  Google Scholar 

  2. B. C. Arnold, N. Balakrishnan, and H. N. Nagaraja, Records (Wiley, New York, 1998).

    Book  Google Scholar 

  3. A. Banerjee, S. Merugu, I. S. Dhillon, and J. Ghosh, ‘‘Clustering with Bregman divergences,’’ J. Mach. Learn. Res. 6 (10), 1705–1749 (2005).

    MathSciNet  MATH  Google Scholar 

  4. M. Basseville, ‘‘Divergence measures for statistical data processing-anannotated bibliography,’’ Signal Processing. 93 (4), 621–633 (2013).

    Article  Google Scholar 

  5. L. M. Bregman, ‘‘The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming,’’ USSR Comput. Math. Math. Phys. 7 (3), 200–217 (1967).

    MathSciNet  Article  Google Scholar 

  6. C. Carota, G. Parmigiani, and N. G. Polson, ‘‘Diagnostic measures for model criticism,’’ J. Am. Stat. Assoc. 91 (434), 753–762 (1996).

    MathSciNet  Article  Google Scholar 

  7. J. H. Cha and J. Mi, ‘‘Some probability functions in reliability and their applications,’’ Nav. Res. Logist. 54 (2), 128–135 (2007).

    MathSciNet  Article  Google Scholar 

  8. T. F. Cox and G. Czanner, ‘‘A practical divergence measure for survival distributions that can be estimated from Kaplan-Meier curves,’’ Stat. Med. 35 (14), 2406–2421 (2016).

    MathSciNet  Article  Google Scholar 

  9. H. A. David and H. N. Nagaraja, Order Statistics, 3rd ed., Wiley, New Jersey, 2003).

    Book  Google Scholar 

  10. A. Di Crescenzo and M. Longobardi, ‘‘A measure of discrimination between past lifetime distributions,’’ Stat. Probab. Lett. 67 (2), 173–182 (2004).

    MathSciNet  Article  Google Scholar 

  11. N. Ebrahimi and S. Kirmani, ‘‘A measure of discrimination between two residual life-time distributions and its applications,’’ Ann. Inst. Stat. Math. 48 (2),257–265 (1996a).

    MathSciNet  Article  Google Scholar 

  12. N. Ebrahimi and S. Kirmani, ‘‘A characterisation of the proportional hazards model through a measure of discrimination between two residual life distributions,’’ Biometrika. 83 (1), 233–235 (1996b).

    MathSciNet  Article  Google Scholar 

  13. A. Fischer, ‘‘Quantization and clustering with Bregman divergences,’’ J. Multivar. Anal. 101 (9), 2207–2221 (2010).

    MathSciNet  Article  Google Scholar 

  14. R. C. Gupta and S. Kirmani, ‘‘Closure and monotonicity properties of nonhomogeneous poisson processes and record values,’’ Probab. Eng. Inf. Sci. 2 (4), 475–484 (1988).

    Article  Google Scholar 

  15. S. Kullback and R. A. Leibler, ‘‘On information and sufficiency,’’ Ann. Math. Stat. 22 (1), 79–86 (1951).

    MathSciNet  Article  Google Scholar 

  16. C. D. Lai and M. Xie, Stochastic ageing and dependence for reliability, Springer Science and Business Media, 2006.

    MATH  Google Scholar 

  17. N. J. Lynn and N. D. Singpurwalla, ‘‘[Burn-in]: Comment: ‘‘burn-in’’ makes us feel good,’’ Stat. Sci. 12 (1), 13–19 (1997).

    Google Scholar 

  18. Z. Mansourvar and M. Asadi, ‘‘An extension of the Cox-Czanner divergence measure to residual lifetime distributions with applications,’’ Statistics. 54 (6), 1311–1328 (2020).

    MathSciNet  Article  Google Scholar 

  19. G. McLachlan and D. Peel, Finite Mixture Models Wiley, 2004.

    MATH  Google Scholar 

  20. N. Misra, J. Francis, and S. Naqvi, ‘‘Some sufficient conditions for relative aging of life distributions,’’ Probab. Eng. Inf. Sci. 31 (1), 83–99 (2017).

    MathSciNet  Article  Google Scholar 

  21. J. Navarro, Y. del Águila, M. A. Sordo, and A. Suárez-Llorens, ‘‘Preservation of reliability classes under the formation of coherent systems,’’ Appl. Stoch. Models Bus. Ind. 30 (4), 444–454 (2014).

    MathSciNet  Article  Google Scholar 

  22. J. Navarro, Y. Del Águila, M. A. Sordo, and A. Suárez-Llorens, ‘‘Preservation of stochastic orders under the formation of generalized distorted distributions. applications to coherent systems,’’ Methodol. Comput. Appl. 18 (2), 529–545 (2016).

    MathSciNet  Article  Google Scholar 

  23. F. Nielsen and R. Nock, ‘‘On the chi square and higher-order chi distances for approximating f-divergences,’’ IEEE Signal Process. Lett. 21 (1), 10–13 (2013).

    Article  Google Scholar 

  24. M. Nikulin, ‘‘Hellinger distance,’’ hazewinkel, michiel, encyclopedia of mathematics,’’ Springer, Berlin, 2001. https://doi.org/10:1361684-1361686.

    Google Scholar 

  25. I. Sason and S. Verdu, ‘‘f-Divergence inequalities,’’ Trans. Inf. Theory 62 (11), 5973–6006 (2016).

    MathSciNet  Article  Google Scholar 

  26. P. Schlattmann, Medical Applications of Finite Mixture Models, Springer, 2009.

    MATH  Google Scholar 

  27. D. Sengupta and J. V. Deshpande, ‘‘Some results on the relative ageing of two life distributions,’’ J. Appl. Probab. 31 (4), 991–1003 (1994).

    MathSciNet  Article  Google Scholar 

  28. M. Shaked and J. G. Shanthikumar, Stochastic orders, Springer Science and Business Media, 2007.

    Book  Google Scholar 

  29. G. C. Tiao and G. E. Box, ‘‘Some comments on ‘‘Bayes’’ estimators,’’ Am. Stat. 27 (1), 12–14 (1973).

    MathSciNet  Google Scholar 

  30. F. Vonta and A. Karagrigoriou, ‘‘Generalized measures of divergence in survival analysis and reliability,’’ J. Appl. Probab. 47 (1), 216–234 (2010).

    MathSciNet  Article  Google Scholar 

Download references

ACKNOWLEDGMENTS

The authors would like to thank the Associate Editor and two reviewers for their constructive comments that greatly improved the paper.

Funding

The authors declare that they have no conflict of interest.

Author information

Authors and Affiliations

  1. Department of Statistics, Faculty of Mathematics and Statistics, University of Isfahan, 81746-73441, Isfahan, Iran

    Zahra Mansourvar & Majid Asadi

Authors
  1. Zahra Mansourvar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Majid Asadi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Zahra Mansourvar or Majid Asadi.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mansourvar, Z., Asadi, M. On a Time Dependent Divergence Measure between Two Residual Lifetime Distributions. Math. Meth. Stat. 29, 135–148 (2020). https://doi.org/10.3103/S1066530720030023

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066530720030023

Keywords:

  • aging properties
  • divergence measures
  • ordered random variables
  • residual lifetime distributions
  • survival transformation model