On the logistic midrange
It is well-known that for a large family of distributions, the sample midrange is asymptotically logistic. In this article, the logistic midrange is closely examined. Its distribution function is derived using Dixon's formula (Bailey (1935, Generalized Hypergeometric Series, Cambridge University Press, p. 13)) for the generalized hypergeometric function with unit argument, together with appropriate techniques for the inversion of (bilateral) Laplace transforms. Several relationships in distribution are established between the midrange and sample median of the logistic and Laplace random variables. Possible applications in testing for outliers are also discussed. © 1987 Kluwer Academic Publishers.
Annals of the Institute of Statistical Mathematics
George, E., & Rousseau, C. (1987). On the logistic midrange. Annals of the Institute of Statistical Mathematics, 39 (1), 627-635. https://doi.org/10.1007/BF02491495