The event-triggered state estimation problem for linear time-invariant systems is considered in the framework of Maximum Likelihood (ML) estimation in this paper. We show that the optimal estimate is parameterized by a special time-varying Riccati equation, and the computational complexity increases exponentially with respect to the time horizon. For ease in implementation, a one-step event-based ML estimation problem is further formulated and solved, and the solution behaves like a Kalman filter with intermittent observations. For the one-step problem, the calculation of upper and lower bounds of the communication rates from the process side is also briefly analyzed. An application example to sensorless event-based estimation of a DC motor system is presented and the benefits of the obtained one-step event-based estimator are demonstrated by comparative simulations. (C) 2013 Elsevier Ltd. All rights reserved.