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Optimal bounds on expectations of order statistics and spacings from nonparametric families of distributions generated by convex transform order

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  • Agnieszka Goroncy
  • Tomasz Rychlik

Abstract

Assume that $$X_1,\ldots , X_n$$ X 1 , … , X n are i.i.d. random variables with a common distribution function $$F$$ F which precedes a fixed distribution function $$W$$ W in the convex transform order. In particular, if $$W$$ W is either uniform or exponential distribution function, then $$F$$ F has increasing density and failure rate, respectively. We present sharp upper bounds on the expectations of single order statistics and spacings based on $$X_1,\ldots , X_n$$ X 1 , … , X n , expressed in terms of the population mean and standard deviation, for the family of all parent distributions preceding $$W$$ W in the convex transform order. We also characterize the distributions which attain the bounds, and specify the general results for the distributions with increasing density function. Copyright Springer-Verlag Berlin Heidelberg 2015

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metrik:v:78:y:2015:i:2:p:175-204
    DOI: 10.1007/s00184-014-0495-7
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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. 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.
    3. 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. J. M. Fernández-Ponce & M. R. Rodríguez-Griñolo, 2017. "New properties of the orthant convex-type stochastic orders," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(3), pages 618-637, September.
    2. Paweł Marcin Kozyra & Tomasz Rychlik, 2017. "Lower and upper bounds on the variances of spacings," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(2), pages 417-428, April.
    3. Rychlik, Tomasz, 2017. "Evaluations of quantiles of system lifetime distributions," European Journal of Operational Research, Elsevier, vol. 256(3), pages 935-944.
    4. 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.

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