Generalized quantile processes
A quantile function pertaining to a probability measure \(P\) defined on the Borel subsets of the \(d\)-dimensional Euclidean space is introduced. This quantile function depends on a subclass \(A\) of all Borel sets as well as a real-valued function \(\lambda\) defined on \(A\). Particular choices of \(A\) and \(\lambda\) yield quantile functions describing various features of \(P\). When the concept is applied to the empirical measure based on independent and identically distributed \(d\)-dimensional random vectors, one obtains generalizations of the well-known one-dimensional empirical quantile process. Functional limit theorems are proved for these generalized quantile processes. Several examples arising from appropriate choices of \(A\) and \(\lambda\) are also discussed.
scientific article; zbMATH DE number 64518