🤖 AI Summary
This study addresses the limitation of the classical Weitzman overlap coefficient, which is restricted to pairwise probability distributions, by extending it for the first time to the case of k (k ≥ 2) independent distributions. The authors reformulate the generalized overlap coefficient as the expectation of a specific function and develop a nonparametric estimator that circumvents the need for closed-form density expressions by integrating kernel density estimation with the method of moments. The resulting framework offers a flexible and broadly applicable measure of overlap among multiple distributions. Extensive Monte Carlo simulations demonstrate that the proposed estimator exhibits robust performance across diverse distributional settings, combining strong theoretical validity with practical utility, thereby providing a valuable tool for multivariate overlap analysis.
📝 Abstract
This papers presents a generalization of the Weitzman overlapping coefficient, originally defined for two probability density functions, to a setting involving k independent distributions, denoted by Delta. To estimate this generalized coefficient, we develop nonparametric methods based on kernel density estimation using k independent random samples (k>=2). Given the analytical complexity of directly deriving Delta using kernel estimators, a novel estimation strategy is proposed. It reformulates Delta as the expected value of a suitably defined function, which is then estimated via the method of moments and the resulting expressions are combined with kernel density estimators to construct the proposed estimators. This method yields multiple new estimators for the generalized Weitzman coefficient. Their performance is evaluated and compared through extensive Monte Carlo simulations. The results demonstrate that the proposed estimators are both effective and practically applicable, providing flexible tools for measuring overlap among multiple distributions.