The problem of parameter estimation by the continuous time observations of a deterministic signal in white Gaussian noise is considered. The asymptotic properties of the maximum likelihood estimator are described in the asymptotic of small noise (large signal-to-noise ratio). We are interested in the situation when there is a misspecification in the regularity conditions. In particular it is supposed that the statistician uses a discontinuous (change-point type) model of signal, when the true signal is continuously differentiable function of the unknown parameter.
"On misspecifications in regularity and properties of estimators." Electron. J. Statist. 12 (1) 80 - 106, 2018. https://doi.org/10.1214/17-EJS1385