On an Inequality for Order Statistics

Abstract

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.