The Overlapping Coefficient: A Decision-Theoretic Study

The overlapping coefficient (OVL) is a widely used measure for quantifying the similarity between probability distributions. Although its relationship with classification error has been studied in previous literature, its role within an overlap-augmented decision criterion merits further examination. This paper examines the connection between distributional overlap and classification error under a decision-theoretic framework. For two probability density functions, f(x) and g(x), the overlapping coefficient is defined as the integral of their pointwise minimum. Under equal prior probabilities and equal misclassification costs, the minimum achievable probability of classification error is one-half of the overlapping coefficient. This is an established result, which serves as a foundation for the present study rather than a new theorem. We also discuss the relationship between the overlapping coefficient and total variation distance, and introduce an overlap-augmented loss function that combines false-positive and false-negative costs with a penalty for distributional overlap. The proposed formulation provides an additional measure for evaluating the difficulty of distinguishing two populations, although the overlap penalty is constant when the underlying distributions are fixed and therefore does not change the optimal classification rule. A numerical example illustrates the relationship among distributional overlap, classification error, and the proposed loss function. Applications to business risk assessment, medical diagnosis, and reliability analysis are discussed.