The Annals of Statistics

On the Joint Asymptotic Distribution of Extreme Midranges

C. Zachary Gilstein

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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

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

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


6275 6215 Midrange extreme order statistic asymptotic distribution maximum likelihood estimate


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

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