Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-23T17:52:52.817Z Has data issue: false hasContentIssue false
  • English
  • Français

On sequences of expected maxima and expected ranges

Part of: Distribution theory - Probability Nonparametric inference Integral transforms, operational calculus

Published online by Cambridge University Press:  30 November 2017

Nickos Papadatos
Nickos Papadatos*
Affiliation:
National and Kapodistrian University of Athens
*
* Postal address: Department of Mathematics, Section of Statistics and Operations Research, National and Kapodistrian University of Athens, Panepistemiopolis, 157 84 Athens, Greece. Email address: npapadat@math.uoa.gr
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate conditions in order to decide whether a given sequence of real numbers represents expected maxima or expected ranges. The main result provides a novel necessary and sufficient condition, relating an expected maxima sequence to a translation of a Bernstein function through its Lévy–Khintchine representation.

Keywords

Expected maximaexpected rangeBernstein functionLévy–Khintchine representationorder statistics

MSC classification

Primary: 62G30: Order statistics; empirical distribution functions 60E05: Distributions
Secondary: 44A60: Moment problems
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 4 , December 2017 , pp. 1144 - 1166
Copyright
Copyright © Applied Probability Trust 2017 

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

Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (1992). A First Course in Order Statistics. John Wiley, New York. Google Scholar
Billingsley, P. (1995). Probability and Measure, 3rd edn. John Wiley, New York. Google Scholar
Charalambides, C. A. (2002). Enumerative Combinatorics. Chapman & Hall/CRC, Boca Raton, FL. Google Scholar
David, H. A. and Nagaraja, H. N. (2003). Order Statistics, 3rd edn. John Wiley, Hoboken, NJ. Google Scholar
Ferguson, T. S. (1996). A Course in Large Sample Theory. Chapman & Hall, London. Google Scholar
Hausdorff, F. (1921). Summationsmethoden und momentfolgen. I. Math. Z. 9, 74–109. Google Scholar
Hill, T. P. and Spruill, M. C. (1994). On the relationship between convergence in distribution and convergence of expected extremes. Proc. Amer. Math. Soc. 121, 1235–1243. (Erratum: 128 (2000), 625–626.) CrossRefGoogle Scholar
Hoeffding, W. (1953). On the distribution of the expected values of the order statistics. Ann. Math. Statist. 24, 93–100. CrossRefGoogle Scholar
Huang, J. S. (1998). Sequences of expectations of maximum-order statistics. Statist. Prob. Lett. 38, 117–123. Google Scholar
Kadane, J. B. (1971). A moment problem for order statistics. Ann. Math. Statist. 42, 745–751. (Correction: Ann. Statist. 1 (1973), 797.) Google Scholar
Kadane, J. B. (1974). A characterization of triangular arrays which are expectations of order statistics. J. Appl. Prob. 11, 413–416. Google Scholar
Kolodynski, S. (2000). A note on the sequence of expected extremes. Statist. Prob. Lett. 47, 295–300. Google Scholar
Lagarias, J. C. (2013). Euler's constant: Euler's work and modern developments. Bull. Amer. Math. Soc. (N.S.) 50, 527–628. Google Scholar
Mallows, C. L. (1973). Bounds on distribution functions in terms of expectations of order-statistics. Ann. Prob. 1, 297–303. Google Scholar
Müntz, C. (1914). Über den approximationsatz von Weierstrass. In H. A. Schwartz Festschrift, Berlin, Google Scholar
Schilling, R. L., Song, R. and Vondraček, Z. (2012). Bernstein Functions: Theory and Applications (De Gruyter Studies Math. 37), 2nd edn. De Gruyter, Berlin. Google Scholar
Shah, B. K. (1970). Note on moments of a logistic order statistics. Ann. Math. Statist. 41, 2150–2152. Google Scholar