The quantile function Q is the functional inverse of the cumulative distribution function, Q[F^-1(x)] = x, and John Tukey, Emmanuel Parzen and, of course, Roger Koenker have called our attention to how useful it is for applications and theory. In fact, the communities that we serve often arrive with a probability p and want the associated data x, as in “What’s the worst drought that I’ll see on my farm in 50 years?”

After a short review of some of the important properties of the quantile function, the talk will show how naturally the function is represented by a differential equation and how this leads to an alternative functional representation of data behavior and likelihood. Techniques of functional data analysis are used to estimate a bivariate function Q(p;t) for data distributed over a continuous index t and where Q is a quantile function for either data or residual distribution for fixed t. An illustration involving rainfall on the Canadian prairies is offered, and a project is described where index t is multivariate and indexes time and all three dimensions of space.