On Constraints in Parameter Estimation and Model Misspecification

Under perfect model specification several deterministic (non-Bayesian) parameter bounds have been established, including the Cramer-Rae, Bhattacharyya, and the Barankin bound; where each is known to apply only to estimators sharing the same mean as a function of the true parameter. This requirement of common mean represents a constraint on the class of estimators. While consideration of model misspecification is an additional complexity, the need for constraints remains a necessary consequence of applying the covariance inequality. These inherent constraints will be examined more closely under misspecification and discussed in detail along with a review of Vuong's original contribution of the misspecified Cramer-Rao bound (MCRB). Recent work derives the same MCRB as Vuong via a different approach, but applicable only to a class of estimators that is more restrictive. An argument is presented herein, however, that broadens this class to include all unbiased estimators of the pseudo-true parameters and strengthens the tie to Vuong's work. Interestingly, an inherent constraint of the covariance inequality, when satisfied by the choice in score function, yields a generalization of the necessary conditions identified by Blyth and Roberts to obtain an inequality of the Cramer-Rae type.