Let $x_1, x_2, \cdots, x_n$ be positive, identically distributed, independent random variables. It is of some statistical interest to study the distribution of $z_n = (x_1 + x_2 + \cdots + x_n)/\max (x_1, x_2, \cdots, x_n)$. In this paper we give its characteristic function and in a few cases its distribution. A limiting distribution of fairly wide applicability is given in the last section.
"On a the Test for Homogeneity and Extreme Values." Ann. Math. Statist. 23 (3) 450 - 456, September, 1952. https://doi.org/10.1214/aoms/1177729390