Statistical Papers

, Volume 60, Issue 6, pp 2001–2012 | Cite as

Bivariate quantile residual life: a characterization theorem and statistical properties

  • M. Shafaei Noughabi
  • M. KayidEmail author
Regular Article


The concept of \(\alpha \)-quantile residual life measure plays an important role in statistics, reliability and life testing. In this investigation, the bivariate \(\alpha \)-quantile residual life measure has been proposed and studied. It has been shown that two suitable bivariate quantile residual life characterize the underlying distribution uniquely. Moreover, some concerned statistical and reliability properties have been proven


Bivariate reliability model Bivariate \(\alpha \)-quantile residual life Bivariate failure rate bivariate life distribution 

Mathematics Subject Classification

60E15 62N05 



The authors are deeply grateful to the Editor-in-Chief and two anonymous referees for their careful detailed remarks, which helped improve both content and presentation of the paper. The support of university of Gonabad for work of the first author under grant No. 93–8 is gratefully acknowledged. The work of the second author was supported by King Saud University, Deanship of Scientific Research, College of Science, Research Center.


  1. Basu AP (1971) Bivariate failure rate. J Am Stat Assoc 66:103–104CrossRefGoogle Scholar
  2. Emura T, Nakatochi M, Matsui S, Michimae H, Rondeau V (2017) Personalized dynamic prediction of death according to tumour progression and high-dimensional genetic factors: Meta-analysis with a joint model. Stat Methods Med Res. doi: 10.1177/0962280216688032 MathSciNetCrossRefGoogle Scholar
  3. Franco-Pereira AM, de Uña-Álvarez J (2013) Estimation of a monotone percentile residual life function under random censorship. Biom J 55:52–67MathSciNetCrossRefGoogle Scholar
  4. Gupta RC (2011) Bivariate odds ratio and association measures. Stat Pap 52:125–138MathSciNetCrossRefGoogle Scholar
  5. Gupta RC, Longford ES (1984) On the determination of a distribution by its median residual life function: A functional equation. J Appl Probab 21:120–128MathSciNetCrossRefGoogle Scholar
  6. Joe H, Proschan F (1984) Percentile residual life functions. Oper Res 32:668–678MathSciNetCrossRefGoogle Scholar
  7. Johnson NL, Kotz S (1973) A vector valued multivariate hazard rate. Bull Int Stat Inst 45:570–574Google Scholar
  8. Johnson NL, Kotz S (1975) A vector multivariate hazard rate. J Multivar Anal 5:53–66MathSciNetCrossRefGoogle Scholar
  9. Kulkarni HV, Rattihalli RN (2002) Nonparametric estimation of a bivariate mean residual life function. J Am Stat Assoc 97:907–917MathSciNetCrossRefGoogle Scholar
  10. Kulkarni HV, Putkure BB (2007) A new variant of the bivariate setting the clock back to zero property. Commun Stat Theory Methods 36:2339–2349MathSciNetCrossRefGoogle Scholar
  11. Lai CD, Xie M (2006) Stochastic aging and dependence for reliability. Springer, New YorkGoogle Scholar
  12. Lillo RE (2005) On the median residual lifetime and its aging properties: a characterization theorem and applications. Nav Res Logist 52:370–380MathSciNetCrossRefGoogle Scholar
  13. Lin GD (2009) On the characterization of life distributions via percentile residual lifetimes. Sankhya Ser A 71:64–72MathSciNetzbMATHGoogle Scholar
  14. Mardia KV (1970) Families of bivariate distributions. Griffin, LondonzbMATHGoogle Scholar
  15. Marshall AW, Olkin I (1967) A multivariate exponential distribution. J Am Stat Assoc 62:291–302MathSciNetCrossRefGoogle Scholar
  16. Nair KR, Nair NU (1989) Bivariate mean residual life. IEEE Trans Reliab 38:362–364CrossRefGoogle Scholar
  17. Navarro J, Ruiz JM, Sandoval CJ (2008) Properties of systems with two exchangeable Pareto components. Stat Pap 49:177–190MathSciNetCrossRefGoogle Scholar
  18. Rojo J, Ghebremichael M (2006) Estimation of two ordered bivariate mean residual life functions. J Multivar Anal 97:431–454MathSciNetCrossRefGoogle Scholar
  19. Roy D (1994) Classification of life distributions in multivariate models. IEEE Trans Reliab 43:224–229CrossRefGoogle Scholar
  20. Shaked M, Shanthikumar JG (1991) Dynamic multivariate mean residual life functions. J Appl Probab 28:613–629MathSciNetCrossRefGoogle Scholar
  21. Shih JH, Emura T (2016) Bivariate dependence measures and bivariate competing risks models under the generalized FGM copula. Stat Pap. First Online: 22 December 2016, 1–18. doi:  10.1007/s00362-016-0865-5 MathSciNetCrossRefGoogle Scholar
  22. Song JK, Cho GY (1995) A note on percentile residual life. Sankhya Ser A 57:333–335MathSciNetzbMATHGoogle Scholar
  23. Wells MT, Yeo KP (1996) Density estimation with bivariate censored data. J Am Stat Assoc 19:1566–1574MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Statistics and Operations Research College of ScienceKing Saud University RiyadhRiyadhSaudi Arabia
  2. 2.Department of Mathematics and StatisticsUniversity of GonabadGonabadIran
  3. 3.Department of Mathematics and Computer ScienceFaculty of Science, Suez UniversitySuezEgypt

Personalised recommendations