The Annals of Statistics

On the Joint Asymptotic Distribution of Extreme Midranges

C. Zachary Gilstein

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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

Permanent link to this document
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
links.jstor.org

Keywords
6275 6215 Midrange extreme order statistic asymptotic distribution maximum likelihood estimate

Citation

Gilstein, C. Zachary. On the Joint Asymptotic Distribution of Extreme Midranges. Ann. Statist. 11 (1983), no. 3, 913--920. doi:10.1214/aos/1176346257. https://projecteuclid.org/euclid.aos/1176346257


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