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Reliability properties of proportional hazards relevation transform

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Abstract

Relevation transform introduced by Krakowski (Rev Fr d’Autom Inf Rech opér Rech opér 7(V2):107–120, 1973) is extensively studied in the literature. In this paper, we study the reliability properties of a special case of relevation transform namely proportional hazards relevation transform. Various stochastic orders and ageing concepts are discussed. A new lifetime distribution called proportional hazards relevated Weibull is introduced and discussed its applications with two real datasets.

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References

  • Barlow RE, Proschan F (1975) Statistical theory of reliability and life lesting. Holt, Rinehart and Winston, New York

    MATH  Google Scholar 

  • Baxter LA (1982) Reliability applications of the relevation transform. Nav Res Logist 29(2):323–330

    Article  MathSciNet  MATH  Google Scholar 

  • Bebbington M, Lai C-D, Zitikis R (2007) A flexible Weibull extension. Reliab Eng Syst Saf 92(6):719–726

    Article  Google Scholar 

  • Belzunce F, Lillo RE, Ruiz J-M, Shaked M (2001) Stochastic comparisons of nonhomogeneous processes. Probab Eng Inf Sci 15(2):199–224

    Article  MathSciNet  MATH  Google Scholar 

  • Chukova S, Dimitrov B, Khalil Z (1993) A characterization of probability distributions similar to the exponential. Can J Stat 21(3):269–276

    Article  MathSciNet  MATH  Google Scholar 

  • Grosswald E, Kotz S, Johnson N (1980) Characterizations of the exponential distribution by relevation-type equations. J Appl Probab 17(3):874–877

    Article  MathSciNet  MATH  Google Scholar 

  • Gupta RC, Kirmani S (1990) The role of weighted distributions in stochastic modeling. Commun Stat Theory Methods 19(9):3147–3162

    Article  MathSciNet  MATH  Google Scholar 

  • Johnson NL, Kotz S (1981) Dependent relevations: time-to-failure under dependence. Am J Math Manag Sci 1(2):155–165

    MathSciNet  MATH  Google Scholar 

  • Kalbfleisch JD, Prentice RL (2011) The statistical analysis of failure time data, vol 360. Wiley, New York

    MATH  Google Scholar 

  • Kapodistria S, Psarrakos G (2012) Some extensions of the residual lifetime and its connection to the cumulative residual entropy. Probab Eng Inf Sci 26(1):129–146

    Article  MathSciNet  MATH  Google Scholar 

  • Kochar SC, Wiens DP (1987) Partial orderings of life distributions with respect to their aging properties. Nav Res Logist 34(6):823–829

    Article  MathSciNet  MATH  Google Scholar 

  • Krakowski M (1973) The relevation transform and a generalization of the gamma distribution function. Rev Fr d’Autom Inf Rech opér Rech Rech opér 7(V2):107–120

    MathSciNet  MATH  Google Scholar 

  • Kuş C (2007) A new lifetime distribution. Comput Stat Data Anal 51(9):4497–4509

    Article  MathSciNet  MATH  Google Scholar 

  • Lai CD, Xie M (2006) Stochastic ageing and dependence for reliability. Springer, Berlin

    MATH  Google Scholar 

  • Lai CD, Zhang L, Xie M (2004) Mean residual life and other properties of Weibull related bathtub shape failure rate distributions. Int J Reliab Qual Saf Eng 11(02):113–132

    Article  Google Scholar 

  • Lau KS, Rao BP (1990) Characterization of the exponential distribution by the relevation transform. J Appl Probab 27(3):726–729

    Article  MathSciNet  MATH  Google Scholar 

  • Lawless JF (2003) Statistical models and methods for lifetime data. Wiley series in probability and statistics, Wiley-Interscience, New York

    MATH  Google Scholar 

  • Loh WY (1984) A new generalization of the class of NBU distributions. IEEE Trans Reliab 33(5):419–422

    Article  MATH  Google Scholar 

  • Marshall AW, Olkin I (1997) A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84(3):641–652

    Article  MathSciNet  MATH  Google Scholar 

  • Nair NU, Sankaran PG, Balakrishnan N (2013) Quantile-based reliability analysis. Springer, New York

    Book  MATH  Google Scholar 

  • Navarro J, Águila Y, Sordo MA, Suárez-Llorens A (2013) Stochastic ordering properties for systems with dependent identically distributed components. Appl Stoch Models Bus Ind 29(3):264–278

    Article  MathSciNet  MATH  Google Scholar 

  • Navarro J, Águila Y, Sordo MA, Suárez-Llorens A (2014) Preservation of reliability classes under the formation of coherent systems. Appl Stoch Models Bus Ind 30(4):444–454

    Article  MathSciNet  Google Scholar 

  • Navarro J, Del Águila Y, Sordo MA, Suárez-Llorens A (2016) Preservation of stochastic orders under the formation of generalized distorted distributions. Applications to coherent systems. Methodol Comput Appl Probab 18(2):529–545

    Article  MathSciNet  MATH  Google Scholar 

  • Psarrakos G, Di Crescenzo A (2018) A residual inaccuracy measure based on the relevation transform. Metrika 81(1):37–59

    Article  MathSciNet  MATH  Google Scholar 

  • Salman Suprawhardana M, Prayoto S (1999) Total time on test plot analysis for mechanical components of the RSG-GAS reactor. Atom Indones 25(2):155–161

    Google Scholar 

  • Sengupta D, Deshpande JV (1994) Some results on the relative ageing of two life distributions. J Appl Probab 31(4):991–1003

    Article  MathSciNet  MATH  Google Scholar 

  • Shaked M, Shanthikumar JG (2007) Stoch Orders. Springer, Berlin

    Book  Google Scholar 

  • Shanthikumar J, Baxter LA (1985) Closure properties of the relevation transform. Nav Res Logist 32(1):185–189

    Article  MathSciNet  MATH  Google Scholar 

  • Sordo MA, Psarrakos G (2017) Stochastic comparisons of interfailure times under a relevation replacement policy. J Appl Probab 54(1):134–145

    Article  MathSciNet  MATH  Google Scholar 

  • Sordo MA, Suárez-Llorens A (2011) Stochastic comparisons of distorted variability measures. Insur Math Econ 49(1):11–17

    Article  MathSciNet  MATH  Google Scholar 

  • Sordo MA, Suárez-Llorens A, Bello AJ (2015) Comparison of conditional distributions in portfolios of dependent risks. Ins Math Econ 61:62–69

    Article  MathSciNet  MATH  Google Scholar 

  • Wang S (1996) Premium calculation by transforming the layer premium density. ASTIN Bull J IAA 26(1):71–92

    Article  MathSciNet  Google Scholar 

  • Xie M, Lai CD (1996) Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function. Reliab Eng Syst Saf 52(1):87–93

    Article  Google Scholar 

Download references

Acknowledgements

We thank the referee and the editor for their constructive comments. The second author is thankful to Kerala State Council for Science Technology and Environment (KSCSTE) for the financial support.

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Authors and Affiliations

  1. Department of Statistics, Cochin University of Science and Technology, Cochin, 682022, Kerala, India

    P. G. Sankaran & M. Dileep Kumar

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  1. P. G. Sankaran
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  2. M. Dileep Kumar
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Correspondence to M. Dileep Kumar.

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Sankaran, P.G., Dileep Kumar, M. Reliability properties of proportional hazards relevation transform. Metrika 82, 441–456 (2019). https://doi.org/10.1007/s00184-018-0681-0

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