Maximum Likelihood Estimator Under a Misspecified Model With High Signal-to-Noise Ratio

It is well known that the maximum-likelihood estimator (MLE) under a misspecified model converges to a well defined limit and it is asymptotically Gaussian as the sample size goes to infinity. In this correspondence, we consider a misspecified model with deterministic signal embedded in Gaussian noise and fully characterize the asymptotic performance of the MLE under this misspecified model with high signal-to-noise (SNR). We see that under some regularity conditions, it converges to a well defined limit and is asymptotically Gaussian with high SNR.