Epidemic Change Model for the Exponential Family

Abstract

We assume that a sequence of independent observations are given from an exponential family. It is hypothesised that the sequence has the same natural parameter θ₀. We would like to test if this natural parameter has been subjected to an epidemic change after an unknown point, for an unknown duration in the sequence. The likelihood ratio statistic for testing such an hypothesis is derived and then it's asymptotic null distribution is derived. We discuss the asymptotic behaviour of the maximum likelihood estimates of the change points. We prove that the asymptotic non-null distribution is of the likelihood ratio statistic is normal. Some special cases of the exponential family are considered in detail.

Key Words:

• Change point
• Exponential family
• Maximum likelihood
• Likelihood ratio test

Acknowledgments

This research is supported in part by a research grant from the Faculty Development Board, University of Wisconsin Oshkosh, Department of Mathematics and Statistics, University of Wisconsin Oshkosh, WI 54901.