non parametric estimation

In this paper, quantile-based dynamic cumulative residual and failure extropy measures are introduced. For a presentation of their applications, first, by using the simulation technique, a suitable estimator is selected to estimate these measures from among different estimators. Then, based on the equality of two extropy measures in terms of order statistics, symmetric continuous distributions are characterized. In this regard, a measure of deviation from symmetry is introduced and how it is applied is expressed in a real example. Also, among the common continuous distributions, the generalized Pareto distribution and as a result the exponential distribution are characterized, and based on the obtained results, the exponentiality criterion  of a distribution is proposed.