Bias reduction in transfer function identification

When one random variable is estimated from another measured random variable through a nonlinear mapping constituting the estimator, then any independent additive noise present in the measured variable creates a bias error in the estimated variable. This occurs even if the added noise has zero mean and symmetric density. This bias error can be computed approximately using the second derivative of the mapping when this mapping is available analytically, and hence a bias-corrected estimate can be constructed. We show that this idea can be extended to the case where the mapping is implicitly defined as the solution of a minimization problem, such as in Maximum Likelihood estimation. We also analyze the effect of this bias correction when applied to the estimation of a first order transfer function at one frequency on the basis of a noisy measurement of that transfer function at some other frequency.