Parameter bounds under misspecified models

A class of parameter bounds emerges as a consequence of the covariance inequality, i.e. Cauchy-Schwarz inequality for expectations. The expectation operator forms an inner product space. Flexibility in the choice of expectation integrand and measure for integration exists, however, to establish a class of parameter bounds under a general form of model misspecifi-cation, i.e. distribution mismatch. The Cramér-Rao bound (CRB) primarily, and secondarily the Barankin, Hammersley-Chapman-Robbins, and Bhattacharyya bounds under misspecification are considered. Huber's sandwich covariance is easily established as the misspecified CRB, and a generalization of the Slepian-Bangs formula under misspecification is provided.