The misspecified Cramér–Rao bound conjecture

Let the assumed probability density function be $f(\mathbf{x};\boldsymbol{\theta})$, satisfying Conditions 1–4, and let $\hat{\boldsymbol{\theta}}(\mathbf{x})$ be any (locally) misspecified unbiased estimator as defined in Definition 1. Let $\boldsymbol{\theta}_*^0$ denote the pseudo-true parameter. Define the mean-square error at $\boldsymbol{\theta}_*^0$ by

$$\operatorname{MSE}\bigl(\hat{\boldsymbol{\theta}}(\mathbf{x}),\boldsymbol{\theta}_*^0\bigr)=\mathbb{E}_p\left[(\hat{\boldsymbol{\theta}}(\mathbf{x})-\boldsymbol{\theta}_*^0)(\hat{\boldsymbol{\theta}}(\mathbf{x})-\boldsymbol{\theta}_*^0)^T\right].$$

Misspecified Cramér–Rao bound conjecture. The mean-square error satisfies

$$\operatorname{MSE}\bigl(\hat{\boldsymbol{\theta}}(\mathbf{x}),\boldsymbol{\theta}_*^0\bigr)\succeq \operatorname{MCRB}(\boldsymbol{\theta}_*^0),$$

where the misspecified Cramér–Rao bound is defined in the source's equation (16). This is presented as a common belief concerning the use of the MCRB as a performance bound under model misspecification; the supplied text does not establish whether the claim is proved or disproved.