, Volume 25, Issue 1, pp 150–169 | Cite as

Stochastic comparisons of generalized mixtures and coherent systems

  • Jorge NavarroEmail author
Original Paper


A distribution function \(F\) is a generalized mixture of the distribution functions \(F_1,\ldots ,F_k\) if \(F=w_1 F_1+\ldots +w_k F_k\), where \(w_1,\ldots ,w_k\) are some real numbers (weights) which should satisfy \(w_1+\ldots +w_k=1\). If all the weights are positive, then we have a classical finite mixture. If some weights are negative, then we have a negative mixture. Negative mixtures appear in different applied probability models (order statistics, estimators, coherent systems, etc.). The conditions to obtain stochastic comparisons of classical (positive) mixtures are well known in the literature. However, for negative mixtures, there are only results for the usual stochastic order. In this paper, conditions for hazard rate and likelihood ratio comparisons of generalized mixtures are obtained. These theoretical results are applied in this paper to study distribution-free comparisons of coherent systems using their representations as generalized mixtures. They can also be applied to other probability models in which the generalized mixtures appear.


Generalized mixtures Stochastic comparisons Coherent systems Hazard rate 

Mathematics Subject Classification

62E10 62N05 



I would like to thank the editors and the anonymous reviewers for several helpful suggestions. This research was supported in part by Ministerio de Economía y Competitividad under Grant MTM2012-34023-FEDER.


  1. Agrawal A, Barlow RE (1984) A survey of network reliability and domination theory. Oper Res 32:478–492CrossRefMathSciNetzbMATHGoogle Scholar
  2. Baggs GE, Nagaraja HN (1996) Reliability properties of order statistics from bivariate exponential distributions. Commun Stat Theory Methods 12:611–631MathSciNetzbMATHGoogle Scholar
  3. Balakrishnan N, Cramer E (2008) Progressive censoring from heterogeneous distributions with applications to robustness. Ann Inst Stat Math 60:151–171CrossRefMathSciNetzbMATHGoogle Scholar
  4. Balakrishnan N, Cramer E (2014) The art of progressive censoring: applications to reliability and quality. Birkhäuser, New YorkCrossRefGoogle Scholar
  5. Balakrishnan N, Xie Q (2007a) Exact inference for a simple step-stress model with type-I hybrid censored data from the exponential distribution. J Stat Plan Inference 137:3268–3290CrossRefMathSciNetzbMATHGoogle Scholar
  6. Balakrishnan N, Xie Q (2007b) Exact inference for a simple step-stress model with type-II hybrid censored data from the exponential distribution. J Stat Plan Inference 137:2543–2563CrossRefMathSciNetzbMATHGoogle Scholar
  7. Balakrishnan N, Xie Q, Kundu D (2009) Exact inference for a simple step-stress model from the exponential distribution under time constraint. Ann Inst Stat Math 61:251–274CrossRefMathSciNetzbMATHGoogle Scholar
  8. Barlow RE, Proschan F (1975) Statistical theory of reliability and life testing. holt, rinehart and winston, New YorkzbMATHGoogle Scholar
  9. Block HW, Li Y, Savits TH (2003) Initial and final behavior of failure rate functions for mixtures and systems. J Appl Probab 40:721–740CrossRefMathSciNetzbMATHGoogle Scholar
  10. Borgonovo E (2010) The reliability importance of components and prime implicants in coherent and non-coherent systems including total-order interactions. Eur J Oper Res 204:485–495CrossRefzbMATHGoogle Scholar
  11. David HA, Nagaraja HN (2003) Order statistics, 3rd edn. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  12. Everitt BS, Hand DJ (1981) Finite mixture distributions. Chapman and Hall, New YorkCrossRefzbMATHGoogle Scholar
  13. Franco M, Balakrishnan N, Kundu D, Vivo JM (2014) Generalized mixtures of weibull components. Test 23:515–535CrossRefMathSciNetzbMATHGoogle Scholar
  14. Glaser RE (1980) Bathtub and related failure rate characterizations. J Am Stat Assoc 75:667–672CrossRefMathSciNetzbMATHGoogle Scholar
  15. Kamps U, Cramer E (2001) On distributions of generalized order statistics. Statistics 35:269–280CrossRefMathSciNetzbMATHGoogle Scholar
  16. Kochar S, Mukerjee H, Samaniego FJ (1999) The “signature” of a coherent system and its application to comparison among systems. Naval Res Logist 46:507–523CrossRefMathSciNetzbMATHGoogle Scholar
  17. Kotz S, Balakrishnan N, Johnson NL (2000) Continuous multivariate distributions. Volume 1: models and applications. Wiley, New YorkCrossRefGoogle Scholar
  18. Navarro J (2008) Likelihood ratio ordering of order statistics, mixtures and systems. J Stat Plan Inference 138:1242–1257CrossRefzbMATHGoogle Scholar
  19. Navarro J, Eryilmaz S (2007) Mean residual lifetimes of consecutive k-out-of-n systems. J App Probab 44:82–98CrossRefMathSciNetzbMATHGoogle Scholar
  20. Navarro J, Balakrishnan N, Samaniego FJ (2008a) Mixture representations of residual lifetimes of used systems. J Appl Probab 45:1097–1112CrossRefMathSciNetzbMATHGoogle Scholar
  21. Navarro J, Guillamon A, Ruiz MC (2009) Generalized mixtures in reliability modelling: applications to the construction of bathtub shaped hazard models and the study of systems. Appl Stoch Models Bus Ind 25:323–337CrossRefMathSciNetzbMATHGoogle Scholar
  22. Navarro J, Pellerey F, Di Crescenzo A (2015) Orderings of coherent systems with randomized dependent components. Eur J Oper Res 240:127–139CrossRefGoogle Scholar
  23. Navarro J, Rubio R (2010) Comparisons of coherent systems using stochastic precedence. Test 19:69–486CrossRefMathSciNetGoogle Scholar
  24. Navarro J, Rubio R (2011) A note on necessary and sufficient conditions for ordering properties of coherent systems with exchangeable components. Naval Res Logist 58:478–489CrossRefMathSciNetzbMATHGoogle Scholar
  25. Navarro J, Rubio R (2012) Comparisons of coherent systems with non-identically distributed components. J Stat Plan Inference 142:1310–1319CrossRefMathSciNetzbMATHGoogle Scholar
  26. Navarro J, Ruiz JM, Sandoval CJ (2007) Properties of coherent systems with dependent components. Commun Stat Theory Methods 36:175–191CrossRefMathSciNetzbMATHGoogle Scholar
  27. Navarro J, Samaniego FJ, Balakrishnan N, Bhattacharya D (2008b) On the application and extension of system signatures in engineering reliability. Naval Res Logist 55:313–327CrossRefMathSciNetzbMATHGoogle Scholar
  28. Navarro J, Shaked M (2006) Hazard rate ordering of order statistics and systems. J Appl Probab 43:391–408CrossRefMathSciNetzbMATHGoogle Scholar
  29. Samaniego FJ (1985) On the IFR closure theorem. IEEE Trans Reliab 34:69–72CrossRefzbMATHGoogle Scholar
  30. Samaniego FJ (2007) System signatures and their applications in engineering reliability. Springer, New YorkCrossRefzbMATHGoogle Scholar
  31. Samaniego FJ, Navarro J (2016) On comparing coherent systems with heterogeneous components. Adv Appl Probab 48(1) (to appear)Google Scholar
  32. Shaked M, Shanthikumar JG (2007) Stochastic orders, 2nd edn. Springer, New YorkCrossRefzbMATHGoogle Scholar
  33. Wu JW (2001) Characterizations of generalized mixtures of geometric and exponential distributions based on upper record values. Stat Papers 42:123–133CrossRefzbMATHGoogle Scholar
  34. Wu JW, Lee WC (1998) Characterization of generalized mixtures of exponential distribution based on conditional expectation of order statistics. J Jpn Stat Soc 28:39–44CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2015

Authors and Affiliations

  1. 1.Facultad de MatemáticasUniversidad de MurciaMurciaSpain

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