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Stochastic comparisons of mixtures of convexly ordered distributions with applications in reliability theory

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  • Belzunce, Félix
  • Shaked, Moshe

Abstract

In this paper we first obtain several results which compare mixtures of distributions in the (increasing) convex and concave stochastic orders. We employ these results to derive relatively weak conditions on the intensity functions of a pair of nonhomogeneous Poisson processes (in fact, on the distribution functions that are associated with these intensity functions) under which the corresponding epoch times of the two nonhomogeneous Poisson processes are ordered in the increasing convex stochastic order. Applications include bounds on the epoch times of a nonhomogeneous Poisson process whose intensity function is the hazard rate function of a new better than used in expectation (new worse than used in expectation) random variable, and the increasing convex ordering of times to the first perfect repair in a Bayesian imperfect repair model.

Suggested Citation

  • Belzunce, Félix & Shaked, Moshe, 2001. "Stochastic comparisons of mixtures of convexly ordered distributions with applications in reliability theory," Statistics & Probability Letters, Elsevier, vol. 53(4), pages 363-372, July.
    • Handle: RePEc:eee:stapro:v:53:y:2001:i:4:p:363-372
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    References listed on IDEAS

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    1. Denuit, Michel, 2000. "Time stochastic s-convexity of claim processes," Insurance: Mathematics and Economics, Elsevier, vol. 26(2-3), pages 203-211, May.
    2. Hesselager, Ole, 1998. "Closure properties of some partial orderings under mixing," Insurance: Mathematics and Economics, Elsevier, vol. 22(2), pages 163-170, June.
    3. Hu, Taizhong & Pan, Xiaoming, 1999. "Preservation of multivariate dependence under multivariate claim models," Insurance: Mathematics and Economics, Elsevier, vol. 25(2), pages 171-179, November.
    4. Sheldon M. Ross & Zvi Schechner, 1984. "Some Reliability Applications of the Variability Ordering," Operations Research, INFORMS, vol. 32(3), pages 679-687, June.
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    Cited by:

    1. Balakrishnan, N. & Kamps, U. & Kateri, M., 2009. "Minimal repair under a step-stress test," Statistics & Probability Letters, Elsevier, vol. 79(13), pages 1548-1558, July.
    2. Khaledi, Baha-Eldin & Shojaei, Roohollah, 2007. "On stochastic orderings between residual record values," Statistics & Probability Letters, Elsevier, vol. 77(13), pages 1467-1472, July.
    3. Badía, Francisco German & Sangüesa, Carmen & Cha, Ji Hwan, 2018. "Stochastic comparisons and multivariate dependence for the epoch times of trend renewal processes," Journal of Multivariate Analysis, Elsevier, vol. 168(C), pages 174-184.
    4. Félix Belzunce & Eva-María Ortega & José Ruiz, 2005. "A note on replacement policy comparisons from NBUC lifetime of the unit," Statistical Papers, Springer, vol. 46(4), pages 509-522, October.
    5. Mohamed Kayid & Mansour Shrahili, 2023. "Characterization Results on Lifetime Distributions by Scaled Reliability Measures Using Completeness Property in Functional Analysis," Mathematics, MDPI, vol. 11(6), pages 1-15, March.
    6. Balakrishnan, Narayanaswamy & Belzunce, Félix & Sordo, Miguel A. & Suárez-Llorens, Alfonso, 2012. "Increasing directionally convex orderings of random vectors having the same copula, and their use in comparing ordered data," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 45-54.
    7. Belzunce, Félix & Mercader, José A. & Ruiz, José M., 2003. "Multivariate aging properties of epoch times of nonhomogeneous processes," Journal of Multivariate Analysis, Elsevier, vol. 84(2), pages 335-350, February.

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