Accurate estimation of impedance functions is essential for the correct interpretation of magnetotelluric (MT) measurements. Noise is inevitably encountered when MT observations are conducted and, consequently, impedance estimates are usually based on least-squares (LS) regression. Least squares ultimately assumes simple Gaussian statistics. However, estimation procedures based on LS would not be statistically optimal, as outliers (abnormal data) are frequently superimposed on a normal ambient MT noise field which is approximately Gaussian. In this situation, the estimation can be seriously misleading.An alternative method for making unbiased robust estimates of MT impedance functions is based on regression M-estimation and the Hilbert Transform, operating on minimum-phase MT impedance functions. In the resulting regression estimates, outlier contamination is removed and other departures from Gauss-Markov optimality are not critical. Using MT data from the Columbia River Plateau and the EMSLAB Lincoln line, it is shown that the method can produce usable MT impedance functions even under conditions of severe noise contamination and in the absence of remote reference data.