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.
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).
Chandler, K.N., “The distribution and frequency of record values,” J. Roy. Statist. Soc., Ser. B 14, 220–228, (1952).
David, H.A., Order statistics (second edition), Wiley, New York, 1981.
Dudewicz, E.J. and van der Meulen, E.C., “Entropy-based tests of uniformity,” J. Amer. Statist. Assoc. 76, 967–974, (1981).
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).
Galambos, J., The asymptotic theory of extreme order statistics (second edition), Krieger Publishing Company, Florida, 1987.
Gumbel, E.J., “The maxima of the mean largest value and range,” Ann. Math. Statist. 5, 76–84, (1954).
Hartley, H.O. and David, H.A., “Universal bounds for mean range and extreme observation,” Ann. Math. Statist. 25, 85–99, (1954).
Kirmani, S.N.U.A. and Alam, S.N., “On goodness of fit tests based on spacings,” Sankhya Ser. A 36, 197–203, (1974).
Moriguti, S., “A modification of Schwarz's inequality with applications to distributions,” Ann. Math. Statist. 24, 107–113, (1953).
Plackett, R.L., “Limits of the ratio of mean range to standard deviation,” Biometrika 34, 120–122, (1947).
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).
Rychlik, T., “Sharp bounds on L-estimates and their expectations for dependent samples,” Commun. Statist.—Theory Meth. 22, 1053–1068, (1993).
Rychlik, T., Projecting statistical functionals, Lectures Notes in Statistics, Volume 160, Springer-Verlag, New York, 2001.
Author information
Authors and Affiliations
Rights 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
Issue Date:
DOI: https://doi.org/10.1023/B:EXTR.0000031182.32627.43