Isotonicity properties of generalized quantiles

We investigate whether several families of generalized quantiles (expectiles, Lp-quantiles and M-quantiles) respect various stochastic orders (the usual stochastic order, the convexity order, and the p-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≤1/2 are basically the only generalized quantiles that are isotonic with respect to the ≤icv ordering; more generally, the Lp-quantiles are isotonic with respect to the p-convex order.