For a sample of independent observations \(X_{1}, X_{2}, \ldots, X_{n}\) with common distribution \(F\) the ordered sample values

\[X_{(1)} \leq X_{(2)} \leq \cdots \leq X_{(n)}\] are called the order statistics.

Todo: connection to maximum processes

Hung Chenâ€™s notes are good.

Gwern did some fun engineering of order statistics, which edges around some general properties of joint maximal statistics of elliptical copulas.

My one cool trick in this domain is for sums of top-\(k\)th of \(N\) i.i.d. exponential random variables, which turn out to have a simple representation in terms of \(k\) random exponentials (Nagaraja 2006). The magic is that quantile transforms make this into a very general way of doing cheap order statistics for i.i.d. variables.

Nagaraja, H. N. 2006. â€śOrder Statistics from Independent Exponential Random Variables and the Sum of the Top Order Statistics.â€ť In *Advances in Distribution Theory, Order Statistics, and Inference*, edited by N. Balakrishnan, JosĂ© MarĂa Sarabia, and Enrique Castillo, 173â€“85. Boston, MA: BirkhĂ¤user Boston. https://doi.org/10.1007/0-8176-4487-3_11.