Detecting change-points in a Compound Poisson Process

Abstract

A statistical change point problem was first studied in the mid-1950s in the context of quality control in industrial processes. A change point is defined as a point in the time order when the probability distribution of a sequence of observations differs before and after that point. The literature of statistical change point has evolved over time and now includes a significant amount of scholarly work on change point analysis with many important applications in other disciplines such as economics, geosciences, medicine, and genetics, to name a few. This work examines the problem of locating changes in the distribution of a Compound Poisson Process where the variables being summed are iid normal and the number of variable follows Poisson. The maximum likelihood ratio for the location of the change point will be explored as well as an information criterion developed, for the case of known variance, while a Bayesian approach is used to deal with the case including change in variance. These results can be applied in any field of study where an interest in locating changes not only in the parameter of a normally distributed data set but also in the rate of their occurrence. It has direct application to the study of gene expression data in cancer research, where it is known that the distances between the genes can affect their expression level.