On CRB for Parameter Estimation in Two Component Gaussian Mixtures and the Impact of Misspecification

A closed form expression for the Cramer Rao lower bound (CRB) for parameter estimation in the presence of a two component Gaussian mixture noise model is derived. It is further shown that this closed form expression can be lower bounded by a simple two term expression. Closed form expressions are also derived for the variance of the maximum likelihood estimator (MLE) when the parameters of the Gaussian mixture model are misspecified. It is then shown that the MLE can handle a significant amount of misspecification of the parameters of the Gaussian mixture model and yet maintain a variance close to the CRB.