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Bounds on the Expectations of \(\boldsymbol{L}\)-Statistics Based on iid Life Distributions

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Abstract

We consider the order statistics based on independent identically distributed non-negative random variables. We determine sharp upper bounds on the expectations of arbitrary linear combinations of order statistics, expressed in the scale units being the \(p\)th roots of \(p\)th raw moments of original variables for various \(p\geq 1\). The bounds are more precisely described for the single order statistics and spacings. The lower bounds are concluded from the upper ones.

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REFERENCES

  1. N. Balakrishnan, ‘‘Improving the Hartley-David-Gumbel bound for the mean of extreme order statistics,’’ Statist. Probab. Lett. 9, 291–294 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  2. N. Balakrishnan, ‘‘A simple application of binomial-negative binomial relationship in the derivation of sharp bounds for moments of order statistics based on greatest convex minorants,’’ Statist. Probab. Lett. 18, 301–305 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Bieniek, ‘‘Projection bounds on expectations of generalized order statistics from DFR and DFRA families,’’ Statistics 40, 339–351, (2006).

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Bieniek, ‘‘Bounds for expectations of differences of generalized order statistics based on general and life distributions,’’ Comm. Statist. Theory Meth. 36, 59–72 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Bieniek, ‘‘Projection bounds on expectations of spacings of generalized order statistics from DD and DDA families,’’ Comm. Statist. Theory Meth. 36, 1343–1357 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Bieniek, ‘‘Projection bounds on expectations of generalized order statistics from DD and DDA families,’’ J. Statist. Plann. Inference 138, 971–981 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Bieniek, ‘‘Projection bounds on expectations of spacings of generalized order statistics from DFR and DFRA families,’’ Statistics 42, 231–243 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Bieniek, ‘‘On families of distributions for which optimal bounds on expectations of GOS can be derived,’’ Comm. Statist. Theory Meth. 37, 1997–2009 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Bieniek, ‘‘Optimal bounds for the mean of the total time on test for distributions with decreasing generalized failure rate,’’ Statistics 50, 1206–1220 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Bieniek and A. Goroncy, ‘‘Sharp lower bounds on expectations of gOS based on DGFR distributions,’’ Statist. Papers 61, 1027–1042 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Bieniek and M. Szpak, ‘‘Sharp bounds for the mean of the total time on test for distributions with increasing generalized failure rate,’’ Statistics 52, 818–828 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  12. K. Danielak, ‘‘Sharp upper mean-variance bounds for trimmed means from restricted families,’’ Statistics 37, 305–324 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  13. K. Danielak and T. Rychlik, ‘‘Sharp bounds for expectations of spacings from DDA and DFRA families,’’ Statist. Probab. Lett. 65, 303–316 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  14. K. Danielak and T. Rychlik, ‘‘Sharp bounds for expectations of spacings from decreasing density and failure rate families,’’ Appl. Math. (Warsaw) 31, 369–395 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  15. L. Gajek and T. Rychlik, ‘‘Projection method for moment bounds on order statistics from restricted families. I. Dependent case,’’ J. Multivariate Anal. 57, 156–174 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  16. L. Gajek and T. Rychlik, ‘‘Projection method for moment bounds on order statistics from restricted families. II. Independent case,’’ J. Multivariate Anal. 64, 156–182 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Goroncy, ‘‘Upper non positive bounds on expectations of generalized order statistics from DD and DDA populations,’’ Comm. Statist. Theory Meth. 46, 11972–11987 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Goroncy, ‘‘On upper bounds on expectations of gOSs based on DFR and DFRA distributions,’’ Statistics 54, 402–414 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Goroncy and T. Rychlik, ‘‘How deviant can you be? The complete solution,’’ Math. Inequal. Appl. 9, 633–647 (2006).

    MathSciNet  MATH  Google Scholar 

  20. A. Goroncy and T. Rychlik, ‘‘Optimal bounds on expectations of order statistics and spacings from nonparametric families of distributions generated by convex transform order,’’ Metrika 78, 175–204 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Goroncy and T. Rychlik, ‘‘Evaluations of expected order statistics and spacings based on the IFR distributions,’’ Metrika 79, 635–657 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Goroncy and T. Rychlik, ‘‘Refined solution to upper bound problem for the expectations of order statistics from decreasing density on the average distributions,’’ Commun. Statist.—Theory Meth. 47, 4029–4041 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  23. E. L. Gumbel, ‘‘The maxima of the mean largest value and of the range,’’ Ann. Math. Statist. 25, 76–84 (1954).

    Article  MathSciNet  MATH  Google Scholar 

  24. H. O. Hartley and H. A. David, ‘‘Universal bounds for the mean range and extreme observation,’’ Ann. Math. Statist. 25, 85–99 (1954).

    Article  MathSciNet  MATH  Google Scholar 

  25. D. S. Mitrinović, Analytic Inequalities, in cooperation with P. M. Vasić, Die Grundlehren der mathematischen Wissenschaften (Springer-Verlag, New York-Berlin, 1970), Vol. 165.

  26. S. Moriguti, ‘‘A modification of the Schwarz’s inequality with applications to distributions,’’ Ann. Math. Statist. 24, 107–113 (1953).

    Article  MathSciNet  MATH  Google Scholar 

  27. N. Papadatos, ‘‘Exact bounds for the expectations of order statistics from non-negative populations,’’ Ann. Inst. Statist. Math. 49, 727–736 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  28. N. Papadatos, ‘‘Optimal moment inequalities for order statistics from nonnegative random variables,’’ Probab. Engrg. Inform. Sci. 35, 316–330 (2021).

    Article  MathSciNet  Google Scholar 

  29. R. L. Plackett, ‘‘Limits of the ratio of mean range to standard deviation,’’ Biometrika 34, 120–122 (1947).

    Article  MathSciNet  MATH  Google Scholar 

  30. T. Rychlik, ‘‘Sharp bounds on \(L\)-estimates and their expectations for dependent samples,’’ Commun. Statist.—Theory Meth. 22, 1053–1068 (1993).

    Article  MATH  Google Scholar 

  31. T. Rychlik, ‘‘Mean-variance bounds for order statistics from dependent DFR, IFR, DFRA, and IFRA samples,’’ J. Statist. Plann. Inference 92, 21–38 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  32. T. Rychlik, ‘‘Optimal mean-variance bounds on order statistics from families determined by star ordering,’’ Appl. Math. (Warsaw) 29, 15–32 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  33. T. Rychlik, ‘‘Non-positive upper bounds on expectations of low rank order statistics from DFR populations,’’ Statistics 43, 53–63 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  34. T. Rychlik, ‘‘Bounds on expectations of small order statistics from decreasing density populations,’’ Metrika 70, 369–381 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  35. T. Rychlik, ‘‘Non-positive upper bounds on expectations of small order statistics from DDA and DFRA populations,’’ Metrika 77, 539–557 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  36. T. Rychlik, ‘‘Maximal expectations of extreme order statistics from increasing density and failure rate populations,’’ Commun. Statist.—Theory Meth. 43, 2199–2213 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  37. T. Rychlik and M. Szymkowiak, ‘‘Bounds on the lifetime expectations of series systems with IFR component lifetimes,’’ Entropy 23 (385), 14 (2021).

    Article  MathSciNet  Google Scholar 

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ACKNOWLEDGMENTS

The author thank the anonymous referees for helpful comments which allowed him to eliminate some mistakes and improving the presentation.

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  1. Institute of Mathematics Polish Academy of Sciences, 00656, Warsaw, Poland

    Tomasz Rychlik

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  1. Tomasz Rychlik
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Rychlik, T. Bounds on the Expectations of \(\boldsymbol{L}\)-Statistics Based on iid Life Distributions. Math. Meth. Stat. 31, 43–56 (2022). https://doi.org/10.3103/S1066530722020041

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