Two new classes of parametric, frequency domain approaches are proposed for estimation of the parameters of scalar, linear "errors-in-variables" models, i.e., linear systems where measurements of both input and output of the system are noise contaminated. One of the proposed classes of approaches consists of linear estimators where using the bispectrum or the integrated polyspectrum (bispectrum or trispectrum) of the input and the cross-bispectrum or the integrated cross-polyspectrum (respectively, of the input-output), the system transfer function is first estimated at a number of frequencies exceeding one-half the number of unknown parameters. The estimated transfer function is then used to estimate the unknown parameters using an overdetermined linear system of equations. In the second class of approaches, quadratic transfer function matching criteria are optimized by using the results of the linear estimators as initial guesses. Both classes of the parameter estimators are shown to be consistent in any measurement noise that has symmetric probability density function when the bispectral approaches are used. The proposed parameter estimators are shown to be consistent in Gaussian measurement noise when trispectral approaches are used. The input to the system need not be a linear process but must have nonvanishing bispectrum or trispectrum. Computer simulation results are presented in support of the proposed approaches. Performance comparisons with several existing approaches based upon computer simulations are also provided.