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).
"On the Joint Asymptotic Distribution of Extreme Midranges." Ann. Statist. 11 (3) 913 - 920, September, 1983. https://doi.org/10.1214/aos/1176346257