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Projection Method for Moment Bounds on Order Statistics from Restricted Families, : II. Independent Case

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  • Gajek, Leslaw
  • Rychlik, Tomasz

Abstract

The method of projection, proposed in Part I, is applied to derive sharp moment bounds for the expectations of order statistics based onindependentsamples from restricted families of distributions. Three families are considered: life distributions with decreasing failure density, decreasing failure rate, and symmetric unimodal ones. The respective bounds are also numerically compared with those for general populations in both the dependent and independent cases.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:64:y:1998:i:2:p:156-182
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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. W. R. van Zwet, 1964. "Convex transformations: A new approach to slcewness and kurtosis," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 18(4), pages 433-441, December.
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    Cited by:

    1. Mariusz Bieniek, 2015. "Optimal evaluations for the bias of trimmed means of $$k$$ k th record values," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 78(4), pages 437-460, May.
    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. "Non-positive upper bounds on expectations of small order statistics from DDA and DFRA populations," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(4), pages 539-557, May.
    4. Okolewski, Andrzej & Rychlik, Tomasz, 2001. "Sharp distribution-free bounds on the bias in estimating quantiles via order statistics," Statistics & Probability Letters, Elsevier, vol. 52(2), pages 207-213, April.
    5. Danielak, Katarzyna & Raqab, Mohammad Z., 2004. "Sharp bounds on expectations of kth record spacings from restricted families," Statistics & Probability Letters, Elsevier, vol. 69(2), pages 175-187, August.
    6. Rychlik, Tomasz, 2009. "Tight evaluations for expectations of small order statistics from symmetric and symmetric unimodal populations," Statistics & Probability Letters, Elsevier, vol. 79(12), pages 1488-1493, June.
    7. Tomasz Rychlik, 2009. "Bounds on expectations of small order statistics from decreasing density populations," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 70(3), pages 369-381, November.
    8. 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.
    9. Mariusz Bieniek & Luiza Pańczyk, 2023. "On the choice of the optimal single order statistic in quantile estimation," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 75(2), pages 303-333, April.
    10. Mariusz Bieniek & Agnieszka Goroncy, 2020. "Sharp lower bounds on expectations of gOS based on DGFR distributions," Statistical Papers, Springer, vol. 61(3), pages 1027-1042, June.
    11. 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.
    12. Tomasz Rychlik, 2001. "Stability of Order Statistics under Dependence," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(4), pages 877-894, December.
    13. Mariusz Bieniek, 2007. "Variation diminishing property of densities of uniform generalized order statistics," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 65(3), pages 297-309, May.

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