In this paper, we consider optimization-based state estimation for general detectable nonlinear systems subject to unknown disturbances. The main contribution is a novel formulation of the cost function and a novel proof technique, which allows us (i) to ensure robust global exponential stability of the estimation error under a suitable exponential detectability condition and (ii) to overcome several of the drawbacks in the existing literature. In particular, we obtain improved estimates for the disturbance gains and the required minimal estimation horizon (which are independent of some maximum a priori disturbance bound), and provide a unified proof technique which can be used for both full information estimation and moving horizon estimation.