This technical note introduces the closed form maximum likelihood estimator for estimating the coefficients of the non-parametric frequency response function from system identification experiments. It is assumed that the experiments consist of repeated pulse excitations and that both the excitation and system response are measured which leads to an error-in-variables setting. Monte Carlo simulations indicate that the estimator achieves efficiency at low signal-to-noise ratios with only few measurements. Comparison with the least-squares estimator shows that better, unbiased results are obtained.