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Characterizations of Arnold and Strauss' and related bivariate exponential models


  • Kotz, Samuel
  • Navarro, Jorge
  • Ruiz, Jose M.


Characterizations of probability distributions is a topic of great popularity in applied probability and reliability literature for over last 30 years. Beside the intrinsic mathematical interest (often related to functional equations) the results in this area are helpful for probabilistic and statistical modelling, especially in engineering and biostatistical problems. A substantial number of characterizations has been devoted to a legion of variants of exponential distributions. The main reliability measures associated with a random vector X are the conditional moment function defined by m[phi](x)=E([phi](X)X[greater-or-equal, slanted]x) (which is equivalent to the mean residual life function e(x)=m[phi](x)-x when [phi](x)=x) and the hazard gradient function h(x)=-[backward difference]logR(x), where R(x) is the reliability (survival) function, R(x)=Pr(X[greater-or-equal, slanted]x), and [backward difference] is the operator . In this paper we study the consequences of a linear relationship between the hazard gradient and the conditional moment functions for continuous bivariate and multivariate distributions. We obtain a general characterization result which is the applied to characterize Arnold and Strauss' bivariate exponential distribution and some related models.

Suggested Citation

  • Kotz, Samuel & Navarro, Jorge & Ruiz, Jose M., 2007. "Characterizations of Arnold and Strauss' and related bivariate exponential models," Journal of Multivariate Analysis, Elsevier, vol. 98(7), pages 1494-1507, August.
  • Handle: RePEc:eee:jmvana:v:98:y:2007:i:7:p:1494-1507

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    References listed on IDEAS

    1. Marshall, Albert W., 1975. "Some comments on the hazard gradient," Stochastic Processes and their Applications, Elsevier, vol. 3(3), pages 293-300, July.
    2. Johnson, N. L. & Kotz, Samuel, 1975. "A vector multivariate hazard rate," Journal of Multivariate Analysis, Elsevier, vol. 5(1), pages 53-66, March.
    3. Shanbhag, D. N. & Kotz, S., 1987. "Some new approaches to multivariate probability distributions," Journal of Multivariate Analysis, Elsevier, vol. 22(2), pages 189-211, August.
    4. Jorge Navarro & Jose Ruiz, 2004. "A characterization of the multivariate normal distribution by using the hazard gradient," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 56(2), pages 361-367, June.
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    Cited by:

    1. Navarro, Jorge, 2008. "Characterizations using the bivariate failure rate function," Statistics & Probability Letters, Elsevier, vol. 78(12), pages 1349-1354, September.
    2. Kim, Bara & Kim, Jeongsim, 2011. "Representation of Downton’s bivariate exponential random vector and its applications," Statistics & Probability Letters, Elsevier, vol. 81(12), pages 1743-1750.
    3. Navarro, Jorge & Sarabia, José María, 2013. "Reliability properties of bivariate conditional proportional hazard rate models," Journal of Multivariate Analysis, Elsevier, vol. 113(C), pages 116-127.
    4. Rajesh, G. & Abdul-Sathar, E.I. & Maya, R., 2015. "Local linear estimation of residual entropy function of conditional distributions," Computational Statistics & Data Analysis, Elsevier, vol. 88(C), pages 1-14.


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