Skip to main content
SpringerLink
Log in

Inequalities for ordered sums

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Summary

Letxi=yi+zi,i=1, ...n, and writex(1)≦...≦x(n), with corresponding notation for the orderedyi andzi. It is shown, for example, that\(x_{(r)} \geqq \mathop {\max }\limits_{i = 1, \cdots ,r} (y_{(i)} + z_{(r + 1 - i)} )\),r=1, ...n. Inequalities are also obtained for convex (or concave) functions of thex(i). The results lead immediately to bounds for the expected values of order statistics in nonstandard situations in terms of simpler expectations. A small numerical example illustrates the method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arnold, B. C. and Groeneveld, R. A. (1979). Bounds on expectations of linear systematic statistics based on dependent samples,Ann. Statist.,7, 220–223. Correction:8, 1402.

    Article  MathSciNet  Google Scholar 

  2. David, H. A., Kennedy, W. J. and Knight, R. D. (1977). Means, variances, and covariances of normal order statistics in the presence of an outlier,Selected Tables in Mathematical Statistics,5, 75–204.

    Google Scholar 

  3. Mallows, C. L. and Richter, D. (1969). Inequalities of Chebyshev type involving conditional expectations,Ann. Math. Statist.,40, 1922–1932.

    Article  MathSciNet  Google Scholar 

  4. Marshall, A. W. and Olkin, I. (1979).Inequalities: Theory of Majorization and Its Applications, Academic Press, New York.

    MATH  Google Scholar 

  5. Nagaraja, H. N. (1981). Some finite sample results for the selection differential,Ann. Inst. Statist. Math.,33, 437–448.

    Article  MathSciNet  Google Scholar 

  6. Smith, N. L. and Tong, Y. L. (1983). Inequalities for functions of order statistics under an additive model,Ann. Inst. Statist. Math.,35, 255–265.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Iowa State University, Iowa, USA

    H. A. David

Authors
  1. H. A. David
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research supported by U.S. Army Research Office.

About this article

Cite this article

David, H.A. Inequalities for ordered sums. Ann Inst Stat Math 38, 551–555 (1986). https://doi.org/10.1007/BF02482542

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02482542

Key words and phrases

Search

Navigation