A Simpson-based estimation approach for the generalized overlapping coefficient of k ≥ 2 normal distributions

The overlapping coefficient is a fundamental measure of similarity between probability distributions. While the case of two distributions has been extensively studied, extending this measure to multiple populations introduces analytical and computational challenges. This paper proposes a unified estimation framework for the generalized overlapping coefficient of a fixed finite number (k≥2) of normal populations. The method combines Simpson's numerical integration rule with plug-in maximum likelihood estimators of the normal parameters. The resulting estimator is shown to be consistent under appropriate regularity conditions. A Monte Carlo simulation study evaluates its finite-sample performance under various overlap scenarios and sample-size configurations. The results demonstrate competitive performance relative to the benchmark estimator, particularly in several low-overlap settings. The proposed methodology provides a simple, flexible, and readily implementable approach for estimating the generalized overlapping coefficient for any fixed finite number of normal populations without explicitly determining all density-intersection points.