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The dilation order, the dispersion order, and orderings of residual lives

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  • Belzunce, F.
  • Pellerey, Franco
  • Ruiz, J. M.
  • Shaked, Moshe

Abstract

One purpose of this paper is to study the relationship of the dilation order ([less-than-or-equals, slant]dil) to two other stochastic orders: the mean residual life order ([less-than-or-equals, slant]mrl) and the increasing convex order ([less-than-or-equals, slant]icx). Regarding these orders, it is already known that X [less-than-or-equals, slant]mrlY => X [less-than-or-equals, slant]icxY. In this paper we show that for non-negative random variables we actually have X [less-than-or-equals, slant]mrlY => X [less-than-or-equals, slant]dilY => X [less-than-or-equals, slant]icxY (the first implication holds under the assumption that at least one of the two underlying random variables satisfies some aging property). Thus, we refine the result of Theorem 3.A.13 in Shaked and Shanthikumar (1994). Another purpose of this paper is to identify conditions under which all the residual lives, that are associated with two random variables X and Y, are ordered according to the dilation or the dispersion orders. Some of these results extend parts (a) and (b) of Theorem 2.B.13 in Shaked and Shanthikumar (1994).

Suggested Citation

  • Belzunce, F. & Pellerey, Franco & Ruiz, J. M. & Shaked, Moshe, 1997. "The dilation order, the dispersion order, and orderings of residual lives," Statistics & Probability Letters, Elsevier, vol. 33(3), pages 263-275, May.
    • Handle: RePEc:eee:stapro:v:33:y:1997:i:3:p:263-275
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    References listed on IDEAS

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    1. Belzunce, F. & Candel, J. & Ruiz, J. M., 1996. "Dispersive orderings and characterization of ageing classes," Statistics & Probability Letters, Elsevier, vol. 28(4), pages 321-327, August.
    2. Bartoszewicz, J., 1987. "A note on dispersive ordering defined by hazard functions," Statistics & Probability Letters, Elsevier, vol. 6(1), pages 13-16, September.
    3. Ebrahimi, Nader & Kirmani, S. N. U. A., 1996. "Some results on ordering of survival functions through uncertainty," Statistics & Probability Letters, Elsevier, vol. 29(2), pages 167-176, August.
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    1. Belzunce, F., 1999. "On a characterization of right spread order by the increasing convex order," Statistics & Probability Letters, Elsevier, vol. 45(2), pages 103-110, November.
    2. Belzunce, Félix & Martínez-Riquelme, Carolina & Ruiz, José M., 2013. "On sufficient conditions for mean residual life and related orders," Computational Statistics & Data Analysis, Elsevier, vol. 61(C), pages 199-210.
    3. Cheung, Ki Ling & Song, Jing-Sheng & Zhang, Yue, 2017. "Cost reduction through operations reversal," European Journal of Operational Research, Elsevier, vol. 259(1), pages 100-112.
    4. Andreas Eberl & Bernhard Klar, 2023. "Stochastic orders and measures of skewness and dispersion based on expectiles," Statistical Papers, Springer, vol. 64(2), pages 509-527, April.
    5. Belzunce, Félix & Ortega, Eva & Ruiz, José M., 1999. "The Laplace order and ordering of residual lives," Statistics & Probability Letters, Elsevier, vol. 42(2), pages 145-156, April.

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