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.
Similar content being viewed by others
References
Barlow RE, Proschan F (1975) Statistical theory of reliability and life lesting. Holt, Rinehart and Winston, New York
Baxter LA (1982) Reliability applications of the relevation transform. Nav Res Logist 29(2):323–330
Bebbington M, Lai C-D, Zitikis R (2007) A flexible Weibull extension. Reliab Eng Syst Saf 92(6):719–726
Belzunce F, Lillo RE, Ruiz J-M, Shaked M (2001) Stochastic comparisons of nonhomogeneous processes. Probab Eng Inf Sci 15(2):199–224
Chukova S, Dimitrov B, Khalil Z (1993) A characterization of probability distributions similar to the exponential. Can J Stat 21(3):269–276
Grosswald E, Kotz S, Johnson N (1980) Characterizations of the exponential distribution by relevation-type equations. J Appl Probab 17(3):874–877
Gupta RC, Kirmani S (1990) The role of weighted distributions in stochastic modeling. Commun Stat Theory Methods 19(9):3147–3162
Johnson NL, Kotz S (1981) Dependent relevations: time-to-failure under dependence. Am J Math Manag Sci 1(2):155–165
Kalbfleisch JD, Prentice RL (2011) The statistical analysis of failure time data, vol 360. Wiley, New York
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
Kochar SC, Wiens DP (1987) Partial orderings of life distributions with respect to their aging properties. Nav Res Logist 34(6):823–829
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
Kuş C (2007) A new lifetime distribution. Comput Stat Data Anal 51(9):4497–4509
Lai CD, Xie M (2006) Stochastic ageing and dependence for reliability. Springer, Berlin
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
Lau KS, Rao BP (1990) Characterization of the exponential distribution by the relevation transform. J Appl Probab 27(3):726–729
Lawless JF (2003) Statistical models and methods for lifetime data. Wiley series in probability and statistics, Wiley-Interscience, New York
Loh WY (1984) A new generalization of the class of NBU distributions. IEEE Trans Reliab 33(5):419–422
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
Nair NU, Sankaran PG, Balakrishnan N (2013) Quantile-based reliability analysis. Springer, New York
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
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
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
Psarrakos G, Di Crescenzo A (2018) A residual inaccuracy measure based on the relevation transform. Metrika 81(1):37–59
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
Sengupta D, Deshpande JV (1994) Some results on the relative ageing of two life distributions. J Appl Probab 31(4):991–1003
Shaked M, Shanthikumar JG (2007) Stoch Orders. Springer, Berlin
Shanthikumar J, Baxter LA (1985) Closure properties of the relevation transform. Nav Res Logist 32(1):185–189
Sordo MA, Psarrakos G (2017) Stochastic comparisons of interfailure times under a relevation replacement policy. J Appl Probab 54(1):134–145
Sordo MA, Suárez-Llorens A (2011) Stochastic comparisons of distorted variability measures. Insur Math Econ 49(1):11–17
Sordo MA, Suárez-Llorens A, Bello AJ (2015) Comparison of conditional distributions in portfolios of dependent risks. Ins Math Econ 61:62–69
Wang S (1996) Premium calculation by transforming the layer premium density. ASTIN Bull J IAA 26(1):71–92
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
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-018-0681-0