Least-squares change-point estimation for the telegraph process observed at discrete times

Abstract

The telegraph process models a random motion with finite velocity and it is usually proposed as an alternative to diffusion models. The process describes the position of a particle moving on the real line, alternatively with constant velocity+v or−v. The changes of direction are governed by a homogeneous Poisson process with rate λ>0. In this paper, we consider a change-point estimation problem for the rate of the underlying Poisson process by means of the least-squares method under the hypothesis of discrete-time sampling. Consistency, rate of convergence and distributional results for the change-point estimator are obtained under both fixed and random sampling. An application to real data is presented.

Keywords:

• discrete observations
• change-point problem
• random times
• telegraph process

Acknowledgements

We thank two anonymous referees for their constructive criticism and fruitful suggestions which also stimulated the analysis of the change point problem under random sampling.

Notes

Although in 18 the least-squares method is applied to problem where the change occurs only on the mean.