Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-12T19:41:17.509Z Has data issue: false hasContentIssue false
  • English
  • Français

OPTIMAL MOMENT INEQUALITIES FOR ORDER STATISTICS FROM NONNEGATIVE RANDOM VARIABLES

Published online by Cambridge University Press:  05 July 2019

Nickos Papadatos Open the ORCID record for Nickos Papadatos [Opens in a new window]
Nickos Papadatos*
Affiliation:
Department of Mathematics, National and Kapodistrian University of Athens, Panepistemiopolis, 157 84 Athens, Greece E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We obtain the best possible upper bounds for the moments of a single-order statistic from independent, nonnegative random variables, in terms of the population mean. The main result covers the independent identically distributed case. Furthermore, the case of the sample minimum for merely independent (not necessarily identically distributed) random variables is treated in detail.

Keywords

nonnegative random variablesoptimal moment boundsorder statisticsreliability systemssample minimum
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 35 , Issue 2 , April 2021 , pp. 316 - 330
Copyright
Copyright © Cambridge University Press 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Ahsanullah, M. & Raqab, M.Z. (2006). Bounds and Characterizations of Record Statistics. Hauppauge, New York: Nova Science Publishers.Google Scholar
2.Arnold, B.C. & Balakrishnan, N. (1989). Relations, Bounds and Approximations for Order Statistics. Lecture Notes in Statist.,vol. 53. New York: Springer.CrossRefGoogle Scholar
3.Arnold, B.C. & Groeneveld, R.A. (1979). Bounds on expectations of linear systematic statistics based on dependent samples. Ann. Statist. 7: 220–223. Correction: 8, 1401.CrossRefGoogle Scholar
4.Arnold, B.C., Balakrishnan, N., & Nagaraja, H.N. (2008). A First Course in Order Statistics. Classic Edition, Philadelphia: SIAM.CrossRefGoogle Scholar
5.Balakrishnan, N. (1990). Improving the Hartley-David-Gumbel bound for the mean of extreme order statistics. Statist. Probab. Lett. 9: 291–294.CrossRefGoogle Scholar
6.Balakrishnan, N. (1993). A simple application of the 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.CrossRefGoogle Scholar
7.Caraux, G. & Gascuel, O. (1992). Bounds on distribution functions of order statistics for dependent variates. Statist. Probab. Lett. 14: 103–105.CrossRefGoogle Scholar
8.David, H.A. (1981). Order Statistics, 2nd ed., New York: Wiley.Google Scholar
9.David, H.A. & Nagaraja, H.N. (2003). Order Statistics, 3rd ed., Hoboken: Wiley.CrossRefGoogle Scholar
10.Gascuel, O. & Caraux, G. (1992). Bounds on expectations of order statistics via extremal dependences. Statist. Probab. Lett. 15: 143–148.CrossRefGoogle Scholar
11.Gumbel, E.J. (1954). The maxima of the mean largest value and of the range. Ann. Math. Statist. 25: 76–84.CrossRefGoogle Scholar
12.Hartley, H.O. & David, H.A. (1954). Universal bounds for mean range and extreme observations. Ann. Math. Statist. 25: 85–99.CrossRefGoogle Scholar
13.Jasiński, K. & Rychlik, T. (2012). Maximum variance of order statistics from symmetric populations revisited. Statistics 47: 422–438.CrossRefGoogle Scholar
14.Jasiński, K. & Rychlik, T. (2016). Inequalities for variances of order statistics originating from urn models. J. App. Probab. 53: 162–173.CrossRefGoogle Scholar
15.Jones, M.C. & Balakrishnan, N. (2002). How are moments and moments of spacings related to distribution functions?. J. Stat. Plann. Inference (C.R. Rao 80th birthday felicitation volume, Part I), 103: 377–390.CrossRefGoogle Scholar
16.Miziula, P. & Navarro, J. (2018). Bounds for the reliability functions of coherent systems with heterogeneous components. Appl. Stochastic Models Business Industry 34: 158–174.CrossRefGoogle Scholar
17.Moriguti, S. (1953). A modification of Schwarz's inequality with applications to distributions. Ann. Math. Statist. 24: 107–113.CrossRefGoogle Scholar
18.Okolewski, A. (2015). Bounds on expectations of L-estimates for maximally and minimally stable samples. Statistics 50: 903–916.CrossRefGoogle Scholar
19.Papadatos, N. (1995). Maximum variance of order statistics. Ann. Inst. Statist. Math. 47: 185–193.CrossRefGoogle Scholar
20.Papadatos, N. (1997). Exact bounds for the expectations of order statistics from non-negative populations. Ann. Inst. Statist. Math. 49: 727–736.CrossRefGoogle Scholar
21.Papadatos, N. (2001a). Expectation bounds on linear estimators from dependent samples. J. Statist. Plann. Inference 93: 17–27.CrossRefGoogle Scholar
22.Papadatos, N. (2001b). Distribution and expectation bounds on order statistics from possibly dependent variates. Statist. Probab. Lett. 54: 21–31.CrossRefGoogle Scholar
23.Papadatos, N. & Rychlik, T. (2004). Bounds on expectations of L-statistics from without replacement samples. J. Statist. Plann. Inference 124: 317–336.CrossRefGoogle Scholar
24.Placket, R.L. (1947). Limits of the ratio of mean range to standard deviation. Biometrika 34: 120–122.CrossRefGoogle Scholar
25.Raqab, M.Z. (2004). Bounds on the expectations of kth record increments. J. Inequal. Pure Appl. Math. 5(4): Article 104, 11 pp.Google Scholar
26.Raqab, M.Z. & Rychlik, T. (2002). Sharp bounds for the mean of the kth record value. Commun. Statist.–Theory Meth. 31: 1927–1937.CrossRefGoogle Scholar
27.Rychlik, T. (1992). Stochastically extremal distributions of order statistics for dependent samples. Statist. Probab. Lett. 13: 337–341.CrossRefGoogle Scholar
28.Rychlik, T. (1993a). Bounds for expectations of L-estimates for dependent samples. Statistics 24: 9–15.CrossRefGoogle Scholar
29.Rychlik, T. (1993b). Sharp bounds on L-estimates and their expectations for dependent samples. Commun. Statist.–Theory Meth. 22: 1053–1068.Correction: 23, 305–306.CrossRefGoogle Scholar
30.Rychlik, T. (1998). Bounds on expectations of L-estimates. In Balakrishnan, N. & Rao, C.R. (eds.), Order Statistics: Theory and Methods. Handbook of Statistics, vol. 16, Amsterdam: North-Holland, pp. 105–145.Google Scholar
31.Rychlik, T. (2001). Projecting Statistical Functionals. Lecture Notes in Statistics,vol. 160, New York: Springer-Verlag.CrossRefGoogle Scholar
32.Rychlik, T. (2008). Extreme variances of order statistics in dependent samples. Statist. Probab. Lett. 78: 1577–1582.CrossRefGoogle Scholar
33.Rychlik, T. (2014). Maximal dispersion of order statistics in dependent samples. Statistics 49: 386–395.CrossRefGoogle Scholar
34.Sen, P.K. (1959). On the moments of sample quantiles. Calcutta Statist. Assoc. Bull. 9: 1–20.CrossRefGoogle Scholar