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

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 lambda > 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.