Misspecified Barankin Type Bound

In statistical signal processing and estimation theory, discrepancies between the true data-generating model and the assumed model can lead to large estimation errors. The misspecified Cramér-Rao bound (MCRB) quantifies the impact of model mismatch on the mean-squared error (MSE), but it is derived for the asymptotic region, where the estimation errors are small. Consequently, it cannot be used to investigate estimation performance in the non-asymptotic region, where estimation errors are large. For instance, the MCRB is not effective in predicting the threshold phenomenon, which is crucial in many signal processing applications. The Barankin bound offers a tighter bound in the non-asymptotic region, and several works have demonstrated its applicability in predicting the threshold phenomenon. However, it is derived for perfectly specified models, where the true data-generating model matches the assumed model. In this work, a misspecified Barankin-type bound that accounts for model mismatch, enabling investigation of the impact of misspecification on the threshold phenomenon, is derived. The Barankin bound and the MCRB emerge as special cases of this lower bound. To illustrate its utility and investigate threshold signal-to-noise ratio phenomenon, we apply the proposed bound to the problem of direction-of-arrival estimation using a sensor array under model misspecification. The modeling errors induce large errors and distort the ambiguity function, a behavior effectively captured by the proposed bound.