  Despite the simplicity and intuitive interpretation of Minimum Mean Squared
Error (MMSE) estimators, their effectiveness in certain scenarios is
questionable. Indeed, minimizing squared errors on average does not provide any
form of stability, as the volatility of the estimation error is left
unconstrained. When this volatility is statistically significant, the
difference between the average and realized performance of the MMSE estimator
can be drastically different. To address this issue, we introduce a new
risk-aware MMSE formulation which trades between mean performance and risk by
explicitly constraining the expected predictive variance of the involved
squared error. We show that, under mild moment boundedness conditions, the
corresponding risk-aware optimal solution can be evaluated explicitly, and has
the form of an appropriately biased nonlinear MMSE estimator. We further
illustrate the effectiveness of our approach via several numerical examples,
which also showcase the advantages of risk-aware MMSE estimation against
risk-neutral MMSE estimation, especially in models involving skewed,
heavy-tailed distributions.
