Title: An Upper Bound for the Distribution Overlap Index and Its Applications. Abstract: The overlap index between two probability distributions has various applications in statistics, machine learning, and other scientific research. However, approximating the overlap index is challenging when the probability distributions are unknown (i.e., distribution-free settings). This paper proposes an easy-to-compute upper bound for the overlap index without requiring any knowledge of the distribution models. We first utilize the bound to find the upper limit for the accuracy of a trained machine learning model when a domain shift exists.  We additionally employ this bound to study the distribution membership classification of given samples. Specifically, we build a novel, distribution-free, computation-efficient, memory-efficient one-class classifier by converting the bound into a confidence score function. The proposed classifier does not need to train any parameters and requires only a small number of in-class samples to be accurate. The classifier shows its efficacy on various datasets and outperforms many state-of-the-art methods in different one-class classification scenarios, including novelty detection, out-of-distribution detection, and backdoor detection.  The obtained results show significant promise toward broadening the applications of overlap-based metrics. 