Skip to main content
SpringerLink
Log in

Evaluating Improvements for Spacings of Order Statistics

  • Published:
Extremes Aims and scope Submit manuscript

Abstract

We evaluate sharp upper bounds for the consecutive spacings of order statistics from an i.i.d. sample, measured in scale units generated by various central absolute moments of the parent distribution. The bounds are based on the projection method combined with the Hölder inequalities. We characterize the probability distributions for which the bounds are attained. We also evaluate the so obtained bounds numerically and compare them with other existing bounds.

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

  • Arnold, B.C., “P-norm bounds on the expectation of the maximum of a possibly dependent sample,” J. Multivariate Analysis 17, 316–332, (1985).

    Google Scholar 

  • Chandler, K.N., “The distribution and frequency of record values,” J. Roy. Statist. Soc., Ser. B 14, 220–228, (1952).

    Google Scholar 

  • David, H.A., Order statistics (second edition), Wiley, New York, 1981.

    Google Scholar 

  • Dudewicz, E.J. and van der Meulen, E.C., “Entropy-based tests of uniformity,” J. Amer. Statist. Assoc. 76, 967–974, (1981).

    Google Scholar 

  • Gajek, L. and Okolewski, A., “Steffensen-type inequalities for order statistics and record statistics,” Annales Univ. Mariae Curie-Sklodowska Lublin-Polonia LI.1(6), 41–59, (1997).

    Google Scholar 

  • Galambos, J., The asymptotic theory of extreme order statistics (second edition), Krieger Publishing Company, Florida, 1987.

    Google Scholar 

  • Gumbel, E.J., “The maxima of the mean largest value and range,” Ann. Math. Statist. 5, 76–84, (1954).

    Google Scholar 

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

    Google Scholar 

  • Kirmani, S.N.U.A. and Alam, S.N., “On goodness of fit tests based on spacings,” Sankhya Ser. A 36, 197–203, (1974).

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  • Raqab, M.Z., “Projection p-norm bounds on the moments of kth record increments,” J. Statist. Plann. Infer. (to appear), (2004).

  • Raqab, M.Z. and Rychlik, T., “Sharp bounds for the mean of the kth record value,” Commun. Statist.—Theory Meth. 31(11), 1927–1938, (2002).

    Google Scholar 

  • Rychlik, T., “Sharp bounds on L-estimates and their expectations for dependent samples,” Commun. Statist.—Theory Meth. 22, 1053–1068, (1993).

    Google Scholar 

  • Rychlik, T., Projecting statistical functionals, Lectures Notes in Statistics, Volume 160, Springer-Verlag, New York, 2001.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Jordan, Amman, 11942, Jordan

    Mohammad Z. Raqab

Authors
  1. Mohammad Z. Raqab
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Raqab, M.Z. Evaluating Improvements for Spacings of Order Statistics. Extremes 6, 259–273 (2003). https://doi.org/10.1023/B:EXTR.0000031182.32627.43

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:EXTR.0000031182.32627.43

Search

Navigation