Distributional Overlap: Simple, Multivariate, Parametric, and Nonparametric Tests for Alienation, Convergence, and General Distributional Difference Issues

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Abstract
This paper proposes a convenient measure of the degree of distributional overlap, both parametric and nonparametric, useful in measuring the degree of Polarization, Alienation, and Convergence. We show the measure is asymptotically normally distributed, making it amenable to inference in consequence. This Overlap measure can be used in the univariate and multivariate framework, and three examples are used to illustrate its use. The nonparametric Overlap Index has two sources of bias, the first being a positive bias induced by the unknown intersection point of the underlying distribution and the second being a negative bias induced by the expectation of cell probabilities being less than the conditional expected values. We show that the inconsistency problem generated by the first bias, prevalent within this class of Goodness of Fit measure, is limited by the number of intersection points of the underlying distributions. A Monte Carlo study was used to examine the biases, and it was found that the latter bias dominates the former. These biases can be diluted by increasing the number of partitions, but prevails asymptotically nonetheless.

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