## The Annals of Statistics

### On the Joint Asymptotic Distribution of Extreme Midranges

C. Zachary Gilstein

#### Abstract

We derive the joint asymptotic distribution of the $k$ midranges formed by averaging the $i$th smallest normalized order statistic with the $i$th largest normalized order statistic, $i = 1, \cdots, k$. We then derive the distribution of the maximum midrange among these $k$ extreme midranges and the limiting distribution of this maximum as $k \rightarrow \infty$. These results imply that, even in infinite samples, different distributions in the class of symmetric, unimodal distributions with tails that die at least as fast as a double exponential distribution may have different maximum likelihood estimates for the location parameter. We also discuss the application of these results to a test of symmetry suggested by Wilk and Gnanadesikan (1968).

#### Article information

Source
Ann. Statist., Volume 11, Number 3 (1983), 913-920.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176346257

Digital Object Identifier
doi:10.1214/aos/1176346257

Mathematical Reviews number (MathSciNet)
MR707941

Zentralblatt MATH identifier
0538.62013

JSTOR