Statistical inference for the Weitzman overlapping coefficient in a family of distributions

A usual problem in applied statistics is the one related to the estimation of the common area under two distributions, which is usually estimated by means of an overlapping coefficient. Particularly, for the Weitzman overlapping coefficient, we found that it is possible to provide a general expression that facilitates making inferences on this coefficient, for many distributions. This expression depends only on two parameters which are actually functions of the parameters of the selected models, among which we can mention the exponential, Weibull, Gumbel, Fréchet and some other distributions that arise under certain transformations of an exponential random variable. The simplicity of our unifying proposal is illustrated considering three well known distributions that have been individually analyzed in statistical literature. To illustrate the performance of the likelihood confidence intervals obtained for this overlapping coefficient, under our proposal, we carried out some simulation studies that yielded adequate coverage frequencies, and just for the sake of comparison we also computed Bootstrap confidence intervals. A real data set is analyzed to exemplify our proposal.