On the efficiency of misspecified Gaussian inference in nonlinear regression: Application to time-delay and Doppler estimation

Nonlinear regression plays a crucial role in various engineering applications. For the sake of mathematical tractability and ease of implementation, most of the existing inference procedures are derived under the assumption of independent and identically distributed (i.i.d.) Gaussian-distributed data. However, real-world situations often deviate from this assumption, with the true data generating process being a correlated, heavy-tailed and non-Gaussian one. The paper aims at providing the Misspecified Cramér–Rao Bound (MCRB) on the Mean Squared Error (MSE) of any unbiased (in a proper sense) estimator of the parameters of a nonlinear regression model derived under the i.i.d. Gaussian assumption in the place of the actual correlated, non-Gaussian data generating process. As a special case, the MCRB for an uncorrelated, i.i.d. Complex Elliptically Symmetric (CES) data generating process under Gaussian assumption is also provided. Consistency and asymptotic normality of the related Mismatched Maximum Likelihood Estimator (MMLE) will be discussed along with its connection with the Nonlinear Least Square Estimator (NLLSE) inherent to the nonlinear regression model. Finally, the derived theoretical findings will be applied in the well-known problem of time-delay and Doppler estimation for GNSS.