  We investigate a reconfigurable intelligent surface (RIS)-aided near-field
localization system with single-antenna user equipment (UE) and base station
(BS) under hardware impairments by considering a practical phase-dependent RIS
amplitude variations model. To analyze the localization performance under the
mismatch between the practical model and the ideal model with unit-amplitude
RIS elements, we employ the misspecified Cram\'{e}r-Rao bound (MCRB). Based on
the MCRB derivation, the lower bound (LB) on the mean-squared error for
estimation of UE position is evaluated and shown to converge to the MCRB at low
signal-to-noise ratios (SNRs). Simulation results indicate more severe
performance degradation due to the model misspecification with increasing SNR.
In addition, the mismatched maximum likelihood (MML) estimator is derived and
found to be tight to the LB in the high SNR regime. Finally, we observe that
the model mismatch can lead to an order-of-magnitude localization performance
loss at high SNRs.
