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

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.

💡 Research Summary

The paper extends the classic Weitzman overlap coefficient, originally defined for two continuous probability density functions as Δ = ∫ min{f₁(x), f₂(x)} dx, to a setting with k ≥ 2 independent distributions. The generalized coefficient is Δ = ∫ min{f₁(x),…,f_k(x)} dx, which quantifies the common area under the point‑wise minimum of all k densities. Direct evaluation of this integral is infeasible in a non‑parametric context, so the authors reformulate Δ as an expectation: for independent random variables X₁,…,X_k drawn from the respective distributions, define a function g(X₁,…,X_k) that equals 1 when the density of a particular sample attains the minimum among all densities and 0 otherwise. Then Δ = E