Open Access
Translator Disclaimer
October, 1971 On an Inequality for Order Statistics
Harold D. Shane
Ann. Math. Statist. 42(5): 1748-1751 (October, 1971). DOI: 10.1214/aoms/1177693175


The problem of finding a Chebyshev type inequality for random variables with unknown or non-existent variance was considered by Z. W. Birnbaum (1970). In this present paper, a statistic, $T$, similar to, but simpler than Birnbaum's, is considered. The statistic is independent of location and scale parameters for families of bell-shaped distributions and so may be considered to be a competitor to Student's $t$. An inequality establishing an upper bound for $P(|T| > \lambda)$ is proved. This bound is considerably smaller than the corresponding bound found by Birnbaum. Finally, an improvement of the latter is offered.


Download Citation

Harold D. Shane. "On an Inequality for Order Statistics." Ann. Math. Statist. 42 (5) 1748 - 1751, October, 1971.


Published: October, 1971
First available in Project Euclid: 27 April 2007

zbMATH: 0292.62036
MathSciNet: MR341739
Digital Object Identifier: 10.1214/aoms/1177693175

Rights: Copyright © 1971 Institute of Mathematical Statistics


Vol.42 • No. 5 • October, 1971
Back to Top