Title: On multivariate and infinite-dimensional quantiles and statistical depth functions
Abstract: For absolutely continuous real random variables, the cumulative distribution function is known to be a strictly increasing function, and the quantile function is defined as its inverse. Minor adjustment to the definition allows us to define quantile functions for other real random variables that may not have strictly increasing cumulative distributions, while retaining all desirable properties. How does one define quantiles in dimensions greater than one? In this overview talk, we will discuss an alternative and equivalent definition of a quantile, and how that definition can generalize to higher dimensions, including many cases where the dimension may be infinite. We will look at some interesting probabilistic and geometric properties of such multivariate quantiles. In one dimension, sample quantiles also allow us to rank and order the observations. A partial equivalent in higher dimensions is the notion of a statistical depth function (or data-depth, as is often commonly called), and our overview will also include discussions of properties and uses of the depth function.