How Robust is the Overlapping Index η ? Kernel and Bandwidth Choices in Simulation and Real Data

The overlapping index η quantifies the proportion of common area shared by two probability density functions and has been proposed as a distribution-free, directly interpretable measure of group difference for use across the social and behavioral sciences. Although η is theoretically independent of any parametric assumption, its empirical computation relies on kernel density estimation (KDE) which reintroduces two analyst-controlled choices: the kernel function and the bandwidth selector. The consequences of these choices for the behavior of ηˆ have not been systematically documented, and they constitute a possible threat to the reproducibility and cross-analyst comparability of the measure.We report a fully crossed Monte Carlo study evaluating four kernel functions – Gaussian, Epanechnikov, triangular, and cosine – and the five bandwidth selectors implemented in base R: two normal-reference rules of thumb, two cross-validation criteria, unbiased and biased, and the Sheather – Jones plug-in, under controlled departures from normality and variance homogeneity. Data were drawn from the skew-normal family. Performance was indexed by relative mean bias (RMB) and root mean square error (RMSE).The choice of KDE specification had a modest but non-negligible effect: bandwidth selection dominated kernel shape, biased cross-validation produced the largest bias, and Silverman’s rule of thumb the smallest. We recommend Silverman’s rule of thumb with a Gaussian or triangular kernel as a defensible default, we suggest that the kernel and bandwidth selector should be reported as a routine part of measurement practice, and caution against cross-validation bandwidths for η.