On Clustering Criteria for Smooth Distributions

We develop a clustering framework for observations from a population with a smooth distribution function and derive its asymptotic properties under a general setup. We propose an intuitive and easily verifiable clustering criterion based on empirical quantiles and prove limit theorems; this paves the way for examining the asymptotic behavior of the point at which the observations are split into clusters. Our results can then be utilized to construct a test for presence of clusters. As an illustration, we apply our results to testing for the presence of jumps in the popular Merton and Kou models for asset pricing.