Skip to main content
SpringerLink
Log in

Maximum variance of order statistics

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

Abstract

Yang (1982,Bull. Inst. Math. Acad. Sinica,10(2), 197–204) proved that the variance of the sample median cannot exceed the population variance. In this paper, the upper bound for the variance of order statistics is derived, and it is shown that this is attained by Bernoulli variates only. The proof is based on Hoeffding's identity for the covariance.

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

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

    Google Scholar 

  • David, H. (1981).Order Statistics, 2nd ed., Wiley, New York.

    Google Scholar 

  • David, H. and Groeneveld, R. (1982). Measures of local variation in a distribution: expected length of spacing and variances of order statistics,Biometrika,69, 227–232.

    Google Scholar 

  • Gajek, L. and Gather, U. (1991). Moment inequalities for order statistics with applications to characterizations of distributions,Metrika,38, 357–367.

    Google Scholar 

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

    Google Scholar 

  • Hartley, H. and David, H. (1954). Universal bounds for mean range and extreme observation,Ann. Math. Statist.,25, 85–99.

    Google Scholar 

  • Lehmann, E. L. (1966). Some concepts of dependence,Ann. Math. Statist.,37, 1137–1153.

    Google Scholar 

  • Lin, G. and Huang, J. (1989). Variances of sample medians,Statist. Probab. Lett.,8, 143–146.

    Google Scholar 

  • Moriguti, S. (1951). Extremal property of extreme value distributions,Ann. Math. Statist.,22, 523–536.

    Google Scholar 

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

    Google Scholar 

  • Papadatos, N. (1994). Intermediate order statistics with applications to nonparametric estimation,Statist. Probab. Lett. (to appear).

  • Papathanasiou, V. (1990). Some characterizations of distributions based on order statistics,Statist. Probab. Lett.,9, 145–147.

    Google Scholar 

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

    Google Scholar 

  • Sugiura, N. (1962). On the orthogonal inverse expansion with an application to the moments of order statistics,Osaka Math. J.,14, 253–263.

    Google Scholar 

  • Székely, G. and Móri, T. (1985). An extremal property of rectangular distributions,Statist. Probab. Lett.,3, 107–109.

    Google Scholar 

  • Terrell, G. (1983). A characterization of rectangular distributions,Ann. Probab.,11, 823–826.

    Google Scholar 

  • Yang, H. (1982). On the variances of median and some other order statistics,Bull. Inst. Math. Acad. Sinica,10(2), 197–204.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Section of Statistics and Operations Research, University of Athens, Panepistemiopolis, 15784, Athens, Greece

    Nickos Papadatos

Authors
  1. Nickos Papadatos
    View author publications

    You can also search for this author in PubMed Google Scholar

About this article

Cite this article

Papadatos, N. Maximum variance of order statistics. Ann Inst Stat Math 47, 185–193 (1995). https://doi.org/10.1007/BF00773423

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00773423

Key words and phrases

Search

Navigation