In recent years, especially in the field of biostatistics, numerous scientific instruments collect functional data. We generally consider a set of functions $x_1,…,x_n$ defined over a common compact subset of the real line.

An order statistic is any function that provides some ordering of our data. For example, given the observations $1,2,1,5,6$ we could define the order as $1,1,2,5,6$ as the ordering with respect to the magnitude of each observation. When analyzing functional data, we’d often like to describe similar order statistics as scalar data. That is, given a set of functional data, we’re interested in providing a meaningful definition for the $\textit{median}$ curve.