Abstract
By combining the Moriguti and Steffensen inequalities, we obtain sharp upper bounds for the expectations of arbitrary linear combinations of order statistics from iid samples. The bounds are expressed in terms of expectations of the left truncated parent distribution and constants that depend only on the coefficients of the linear combination. We also present analogous results for dependent id samples. The bounds are especially useful for L-estimates of the scale parameter of the distribution.
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References
Arnold BC, Balakrishnan N (1989) Relations, Bounds and Approximations for Order Statistics, Lecture Notes in Statistics, – vol 53 Springer-Verlag, New York
Balakrishnan N (1993) A simple application of binomial-negative binomial relationship in the derivation of sharp bounds for moments of order statistics based on greatest convex minorants. Statist Probab Lett 18:301–305
Balakrishnan N, Cramer E, Kamps U (2001) Bounds for means and variances of progressive type II censored order statistics. Statist Probab Lett 54:301–315
Cramer E, Kamps U, Rychlik T (2002) Evaluations of expected generalized order statistics in various scale units. Appl Math (Warsaw) 29:285–295
Cramer E, Kamps U, Rychlik T (2004) Unimodality of uniform generalized order statistics, with applications to mean bounds. Ann Inst Math Statist 56:183–192
Danielak K, Rychlik T (2003) Exact bounds for the bias of trimmed means. Austral New Zeal J Statist 45:83–96
Gajek L, Okolewski A (1997) Steffensen-type inequalities for order and record statistics. Ann Univ Mariae Curie-Skłodowska Sect A 41:41–59
Gajek L, Okolewski A (2000a) Sharp bounds on moments of generalized order statistics. Metrika 52:27–43
Gajek L, Okolewski A (2000b) Inequalities for generalized order statistics from some restricted family of distributions. Commun Statist—Theory Meth 29:2427–2438
Gajek L, Okolewski A (2001) Improved Steffensen type bounds on expectations of record statistics. Statist Probab Lett 55:205–212
Huang JS (1997) Sharp bounds for the expected value of order statistics. Statist Probab Lett 33:105–107
Ludwig O (1973) Differenzen der Erwarungswerte von Ranggrössen in klienen Stichproben. In: Bereanu B, Iosifescu M, Postelnicu T, Tăutu P (eds) Proc. 4th Conf. on Probab. Theory. Editura Acad pp 299–303
Mitrinovič DS (1970) Analytic Inequalities. Springer-Verlag, Berlin
Moriguti S (1953) A modification of Schwarz’s inequality, with applications to distributions. Ann Math Statist 24:107–113
Raqab MZ (1997) Bounds based on greatest convex minorants for moments of record values. Statist Probab Lett 36:35–41
Raqab MZ, Rychlik T (2002) Sharp bounds for the mean of the kth record value. Commun Statist—Theory Meth 31:1927–1937
Rychlik T (1993) Bounds for expectations of L-estimates for dependent samples. Statistics 24:1–7
Rychlik T (1998). Bounds for expectations of L-estimates. In: Balakrishnan N, Rao CR (eds). Order Statistics : Theory & Methods Handbook of Statistics. vol. 16. North-Holland, Amsterdam, pp. 105–145
Rychlik T (2001) Projecting Statistical Functionals. Lecture Notes in Statistics, vol. 160. Springer-Verlag, New York
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Balakrishnan, N., Rychlik, T. Evaluating expectations of L-statistics by the Steffensen inequality. Metrika 63, 371–384 (2006). https://doi.org/10.1007/s00184-005-0026-7
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DOI: https://doi.org/10.1007/s00184-005-0026-7