Statistical Papers

, Volume 60, Issue 1, pp 223–237 | Cite as

On the unimodality of the likelihood ratio with applications

  • Félix BelzunceEmail author
  • Carolina Martínez-Riquelme
Regular Article


Along this paper, we study the relationship between the unimodality of the likelihood ratio and the behaviour of the ratio of two survival functions. The results provide sufficient conditions for both the hazard rated order when the likelihood ratio order does not hold and the mean residual life order when the hazard rate order does not hold. Examples where these results can be applied are also given.


Likelihood ratio order Hazard rate order Mean residual life order Unimodality 

Mathematics Subject Classification




The authors want to acknowledge the comments by two anonymous referees that have greatly improved the presentation and the contents of this paper. The authors also want to acknowledge the support received from the Ministerio de Economía y Competitividad under Grant MTM2012-34023-FEDER.


  1. Ahmad IA, Kayid M (2005) Characterizations of the RHR and MIT orderings and the DRHR and IMIT classes of life distributions. Probab Eng Inf Sci 19:447–461MathSciNetzbMATHGoogle Scholar
  2. Ahmad IA, Kayid M, Pellerey F (2005) Further results involving the MIT order and IMIT class. Probab Eng Inf Sci 19:377–395MathSciNetzbMATHGoogle Scholar
  3. Ahmed AHN (1988) Preservation properties for the mean residual life ordering. Stat Pap 29:143–150MathSciNetCrossRefzbMATHGoogle Scholar
  4. Alzaid AA (1988a) Mean residual life ordering. Stat Pap 29:35–43MathSciNetCrossRefzbMATHGoogle Scholar
  5. Alzaid AA (1988b) Erratum: “mean residual life ordering”. Stat Pap 30:184CrossRefzbMATHGoogle Scholar
  6. Belzunce F, Martínez-Riquelme C, Mulero J (2015) An introduction to stochastic orders. Elsevier-Academic Press, LondonzbMATHGoogle Scholar
  7. Belzunce F, Martínez-Riquelme C, Ruiz JM (2013) On sufficient conditions for mean residual life and related orders. Comput Stat Data Anal 61:199–210MathSciNetCrossRefzbMATHGoogle Scholar
  8. Cheng MY, Qiu P, Tan X, Tu D (2009) Confidence intervals for the first crossing point of two hazard functions. Lifetime Data Anal 15:441–454MathSciNetCrossRefzbMATHGoogle Scholar
  9. Denuit M, Dhaene J, Goovaerts M, Kaas R (2005) Actuarial theory for dependent risks. Measures, orders and models. Wiley, ChichesterCrossRefGoogle Scholar
  10. Kayid M (2011) Preservation properties of the moment generating function ordering of residual lives. Stat Pap 52:523–529MathSciNetCrossRefzbMATHGoogle Scholar
  11. Kayid M, Ahmad IA (2004) On the mean inactivity time ordering with reliability applications. Probab Eng Inf Sci 18:395–409MathSciNetCrossRefzbMATHGoogle Scholar
  12. Khaledi BE, Amiri L (2011) On the mean residual life order of convolutions of independent uniform random variables. J Stat Plan Inference 141:3716–3724MathSciNetCrossRefzbMATHGoogle Scholar
  13. Lisek B (1978) Comparability of special distributions. Math Oper Stat Ser Stat 9:537–593MathSciNetzbMATHGoogle Scholar
  14. Liu K, Qiu P, Sheng J (2007) Comparing two crossing hazard rates by Cox proportional hazards modelling. Stat Med 26:375–391MathSciNetCrossRefGoogle Scholar
  15. Mantel N, Stablein DM (1988) The crossing hazard function problem. Statistician 37:59–64CrossRefGoogle Scholar
  16. Marshall A, Olkin I, Arnold BC (2011) Inequalities: theory of majorization and its applications, 2nd edn. Springer Series in Statistics. Springer, New YorkGoogle Scholar
  17. Metzger C, Rüschendorf L (1991) Conditional variability ordering of distributions. Ann Oper Res 32:127–140MathSciNetCrossRefzbMATHGoogle Scholar
  18. Müller A, Stoyan D (2002) Comparison methods for stochastic models and risks. Wiley Series in Probability and Statistics. Wiley, ChichesterGoogle Scholar
  19. Nekoukhou V, Alamatsaz MH (2012) A family of skew-symmetric-Laplace distributions. Stat Pap 53:685–696MathSciNetCrossRefzbMATHGoogle Scholar
  20. O’Quigley J (1994) On a two-sided test for crossing hazards. Stat J Inst Stat 43:563–569Google Scholar
  21. Ross SM (1996) Stochastic processes, 2nd edn. Wiley Series in Probability and Statistics. Probability and Statistics. Wiley, New YorkGoogle Scholar
  22. Shaked M, Shanthikumar GJ (2007) Stochastic orders. Springer Series in Statistics. Springer, New YorkGoogle Scholar
  23. Taylor JM (1983) Comparisons of certain distributions functions. Math Oper Stat Ser Stat 14:307–408MathSciNetGoogle Scholar
  24. Wang B, Wang W (2011) Stochastic ordering of folded normal random variables. Stat Probab Lett 81:524–528MathSciNetCrossRefzbMATHGoogle Scholar
  25. Zhao P, Li X (2014) Ordering properties of convolutions from heterogeneous populations: a review on some recent developments. Commun Stat Theory Methods 43:2260–2273MathSciNetCrossRefzbMATHGoogle Scholar
  26. Zhang J, Peng Y (2009) Crossing hazard functions in common survival models. Stat Probab Lett 79:2124–2130MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Dpto. Estadística e Investigación Operativa, Facultad de MatemáticasUniversidad de MurciaMurciaSpain

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