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September, 1983 On the Joint Asymptotic Distribution of Extreme Midranges
C. Zachary Gilstein
Ann. Statist. 11(3): 913-920 (September, 1983). DOI: 10.1214/aos/1176346257


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).


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C. Zachary Gilstein. "On the Joint Asymptotic Distribution of Extreme Midranges." Ann. Statist. 11 (3) 913 - 920, September, 1983.


Published: September, 1983
First available in Project Euclid: 12 April 2007

zbMATH: 0538.62013
MathSciNet: MR707941
Digital Object Identifier: 10.1214/aos/1176346257

Rights: Copyright © 1983 Institute of Mathematical Statistics


Vol.11 • No. 3 • September, 1983
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