Beyond Pairwise: Nonparametric Kernel Estimators for a Generalized Weitzman Coefficient Across k Distributions

TL;DR

This paper introduces nonparametric kernel-based estimators for a generalized Weitzman coefficient that measures overlap among multiple distributions, extending the original two-distribution version.

Contribution

It proposes a novel estimation strategy for the generalized Weitzman coefficient using kernel density estimation and the method of moments for multiple distributions.

Findings

Estimators are effective in simulation studies.

Proposed methods are practically applicable.

Flexible tools for multi-distribution overlap measurement.

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.