A closed-form estimator of R = P(X < Y) based on ranked set sampling for a family of statistical distributions with application in agriculture

In this article, we will derive a closed-form estimator for the probability R = P (X < Y) based on a ranked set sampling (RSS) scheme when the he random variables X and Y are assumed to follow the Lehmann Type-II (L-II) family of distributions. Estimating R through the maximum likelihood (ML) method within the RSS framework does not yield an analytical solution because of the non-linear components present in the likelihood equations. In this context, we employ a modified maximum likelihood (MML) estimation approach to derive a closed-form estimator for R. Estimates of R under both ML and MML techniques along with their corresponding asymptotic confidence intervals are determined and compared in a simulation study under one of the distributions of the L-II family called the inverse Topp-Leone distribution. At the end, the simulation results are strengthened using a real example in the field of agriculture.