  Purpose: To develop neural network (NN)-based quantitative MRI parameter
estimators with minimal bias and a variance close to the Cram\'er-Rao bound.
  Theory and Methods: We generalize the mean squared error loss to control the
bias and variance of the NN's estimates, which involves averaging over multiple
noise realizations of the same measurements during training. Bias and variance
properties of the resulting NNs are studied for two neuroimaging applications.
  Results: In simulations, the proposed strategy reduces the estimates' bias
throughout parameter space and achieves a variance close to the Cram\'er-Rao
bound. In vivo, we observe good concordance between parameter maps estimated
with the proposed NNs and traditional estimators, such as non-linear
least-squares fitting, while state-of-the-art NNs show larger deviations.
  Conclusion: The proposed NNs have greatly reduced bias compared to those
trained using the mean squared error and offer significantly improved
computational efficiency over traditional estimators with comparable or better
accuracy.
