Conditional quantiles: An operator-theoretical approach

Abstract

This paper derives several novel properties of conditional quantiles viewed as nonlinear operators. The results are organized in parallel to the usual properties of the expectation operator. We first define a τ-conditional quantile random set, relative to any sigma-algebra, as a set of solutions of an optimization problem. Then, well-known properties of unconditional quantiles, as translation invariance, comonotonicity, and equivariance to monotone transformations, are generalized to the conditional case. Moreover, a simple proof for Jensen’s inequality for conditional quantiles is provided. We also investigate continuity of conditional quantiles as operators with respect to different topologies and obtain a novel Fatou’s lemma for quantiles. Conditions for continuity in ${\mathsf{L}}^p$ and weak continuity are also derived. Then, the differentiability properties of quantiles are addressed. We demonstrate the validity of Leibniz’s rule for conditional quantiles for the cases of monotone, as well as separable functions. Finally, although the law of iterated quantiles does not hold in general, we characterize the maximum set of random variables for which this law holds, and investigate its consequences for the infinite composition of conditional quantiles.