On Clustering Criteria for Smooth Distributions

We develop a clustering framework, motivated by the problem of testing for jumps in continuous-time stochastic process models, and derive its asymptotic properties under a general setup. Our technique is applicable whenever we have data from a population with a smooth distribution function. We then propose an intuitive and easily verifiable clustering criterion, based on the Empirical Cross-over Function, which provides us with the requisite tools to develop a test for the presence of jumps. We illustrate the validity of our theory on the popular Merton and Kou models for asset pricing with the objective of investigating jumps occurring in these models as a phenomena which leads to the formation of clusters.