Estimation for the discretely observed telegraph process

The telegraph process $\{X(t), t>0\}$ is supposed to be observed at $n+1$ equidistant time points $t_i=i\Delta _n$, $i=0,1,\dots , n$. The unknown value of $\lambda$, the underlying rate of the Poisson process, is a parameter to be estimated. The asymptotic framework considered is the following: $\Delta _n \to 0$, $n\Delta _n=T \to \infty$ as $n \to \infty$. We show that previously proposed moment type estimators are consistent and asymptotically normal but not efficient. We study further an approximated moment type estimator which is still not efficient but comes in explicit form. For this estimator the additional assumption $n\Delta _n^3 \to 0$ is required in order to obtain asymptotic normality. Finally, we propose a new estimator which is consistent, asymptotically normal and asymptotically efficient under no additional hypotheses.