aequationes mathematicae

, Volume 14, Issue 3, pp 293–301 | Cite as

On bounds for the range of ordered variates II

  • Paul R. Beesack
Research papers

Abstract

Letx1, ⋯,x n be real numbers with 1 n x j =0, |x 1 |≦|x 2 |≦⋯≦|x n |, and ∑ 1 n f(|x i |)=A>0, wheref is a continuous, strictly increasing function on [0, ∞) withf(0)=0. Using a generalized Chebycheff inequality (or directly) it is easy to see that an upper bound for |x m | isf −1 (A/(n−m+1)). If (n−m+1) is even, this bound is best possible, but not otherwise. Best upper bounds are obtained in case (n−m+1) is odd provided either (i)f is strictly convex on [0, ∞), or (ii)f is strictly concave on [0, ∞). Explicit best bounds are given as examples of (i) and (ii), namely the casesf(x)=x p forp>1 and 0<p<1 respectively.

AMS (1970) subject classification

Primary 62G30, 26A51 Secondary 26A87 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Beesack, P. R.,On bounds for the range of ordered variates. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz.412–460 (1973), 93–96.Google Scholar
  2. [2]
    Boyd, A. V.,Bounds for order statistics. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz.357–380 (1971), 31–32.Google Scholar
  3. [3]
    Hawkins, D. M.,On the bounds of the range of order statistics. J. Amer. Statist. Assoc.66 (1971), 644–645.Google Scholar
  4. [4]
    Loève, M.,Probability theory,2nd ed. Van Nostrand, New York, 1960.Google Scholar
  5. [5]
    Pearson, E. S. andChandra Sekar, C.,The efficiency of statistical tools and a criterion for the rejection of outlying observations. Biometrika28 (1936), 308–320.Google Scholar
  6. 6]
    Samuelson, P. A.,How deviant can you be? J. Amer. Statist. Assoc.63 (1968), 1522–1525.Google Scholar

Copyright information

© Birkhäuser Verlag 1976

Authors and Affiliations

  • Paul R. Beesack
    • 1
  1. 1.Department of MathematicsCarleton UniversityOttawa 1Canada

Personalised recommendations