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Evaluations of expectations of order statistics and spacings based on IFR distributions

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  • Agnieszka Goroncy

    (Nicolaus Copernicus University)

  • Tomasz Rychlik

    (Polish Academy of Sciences)

Abstract

We consider i.i.d. random variables $$X_1,\ldots , X_n$$ X 1 , … , X n with a distribution function F preceding the exponential distribution function V in the convex transform order which means that F has an increasing failure rate. We determine sharp upper bounds on the expectations of order statistics and spacings based on $$X_1,\ldots , X_n$$ X 1 , … , X n , expressed in the population standard deviation units. We also specify the distributions for which all these bounds are attained. Finally, we indicate some reliability applications.

Suggested Citation

  • Agnieszka Goroncy & Tomasz Rychlik, 2016. "Evaluations of expectations of order statistics and spacings based on IFR distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 79(6), pages 635-657, August.
    • Handle: RePEc:spr:metrik:v:79:y:2016:i:6:d:10.1007_s00184-015-0570-8
    DOI: 10.1007/s00184-015-0570-8
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    References listed on IDEAS

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    1. Gajek, Leslaw & Rychlik, Tomasz, 1996. "Projection Method for Moment Bounds on Order Statistics from Restricted Families: I. Dependent Case," Journal of Multivariate Analysis, Elsevier, vol. 57(1), pages 156-174, April.
    2. Danielak, Katarzyna & Rychlik, Tomasz, 2003. "Sharp bounds for expectations of spacings from DDA and DFRA families," Statistics & Probability Letters, Elsevier, vol. 65(4), pages 303-316, December.
    3. Tomasz Rychlik, 2014. "Maximal Expectations of Extreme Order Statistics from Increasing Density and Failure Rate Populations," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 43(10-12), pages 2199-2213, May.
    4. Agnieszka Goroncy & Tomasz Rychlik, 2015. "Optimal bounds on expectations of order statistics and spacings from nonparametric families of distributions generated by convex transform order," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 78(2), pages 175-204, February.
    5. Gajek, Leslaw & Rychlik, Tomasz, 1998. "Projection Method for Moment Bounds on Order Statistics from Restricted Families, : II. Independent Case," Journal of Multivariate Analysis, Elsevier, vol. 64(2), pages 156-182, February.
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    Cited by:

    1. Rychlik, Tomasz, 2017. "Evaluations of quantiles of system lifetime distributions," European Journal of Operational Research, Elsevier, vol. 256(3), pages 935-944.

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