The estimation of Wasserstein and Zolotarev distances to the class of exponential variables

Given a positive random variable X, we are interested in measuring how well the exponential distribution with the same mean approximates the probability distribution of X, based on the information provided by a sample from X. Specifically, we consider the problem of estimating the Zolotarev distance of order r, r=1,2,..., between X and the exponential distribution with mean E(X). We give sharp results on the asymptotic distribution of a plug-in estimator of this metric and compare it with the finite-sample distribution via simulations. The practical use of the Zolotarev metrics is illustrated analysing a massive data set from X-ray astronomy.