On the asymptotic shape of quantile surfaces
Florian Gach, Simon Hochgerner
Abstract
This article is concerned with the asymptotic shape of quantile surfaces, defined as the set of quantiles at a given level $α$ generated by a controlled one-dimensional distribution. Specifically, when the distribution arises as a linear combination of log-normal random variables and the control is a vector of positive coefficients, we prove that quantile surfaces are globally concave in the left tail ($α\to0$) and globally convex in the right tail ($α\to1$). Moreover, these surfaces exhibit asymptotic separation of scale and shape.
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