Lp-metric under the location-independent risk ordering of random variables

The Lp-metric Δh,p(X) between the survival function F¯ of a random variable X and its distortion h∘F¯ is a characteristic of the variability of X. In this paper, it is shown that if a random variable X is larger than another random variable Y in the location-independent risk order or in the excess wealth order, then Δh,p(X)≥Δh,p(Y) whenever p∈(0,1] and the distortion function h is convex or concave. An alternative and simple proof of the corresponding known result in the literature for the dispersive order is given. Some applications are also presented.