New developments on the Lp-metric between a probability distribution and its distortion

Let Δh,p(X) denote the Lp-metric between the survival function F¯ of a random variable X and its distortion h∘F¯. It is shown that if X is smaller than Y in the convex order, then Δh,p(X)≤Δh,p(Y) whenever p∈(0,1] and the distortion function h is convex or concave. The corresponding results for some other stochastic orders are also presented.