Isotonicity properties of generalized quantiles

We investigate whether several families of generalized quantiles (expectiles, LpLp-quantiles and MM-quantiles) respect various stochastic orders (the usual stochastic order, the convexity order, and the pp-convexity orders).We employ techniques from monotone comparative statics developed by Topkis (1978) and Milgrom and Shannon (1994), in order to provide sufficient as well as necessary conditions for isotonicity.We show that expectiles with α≤1/2α≤1/2 are basically the only generalized quantiles that are isotonic with respect to the ≤icv≤icv ordering; more generally, the LpLp-quantiles are isotonic with respect to the pp-convex order.