  We present a new uncertainty principle for risk-aware statistical estimation,
effectively quantifying the inherent trade-off between mean squared error
($\mse$) and risk, the latter measured by the associated average predictive
squared error variance ($\sev$), for every admissible estimator of choice. Our
uncertainty principle has a familiar form and resembles fundamental and
classical results arising in several other areas, such as the Heisenberg
principle in statistical and quantum mechanics, and the Gabor limit (time-scale
trade-offs) in harmonic analysis. In particular, we prove that, provided a
joint generative model of states and observables, the product between $\mse$
and $\sev$ is bounded from below by a computable model-dependent constant,
which is explicitly related to the Pareto frontier of a recently studied
$\sev$-constrained minimum $\mse$ (MMSE) estimation problem. Further, we show
that the aforementioned constant is inherently connected to an intuitive new
and rigorously topologically grounded statistical measure of distribution
skewness in multiple dimensions, consistent with Pearson's moment coefficient
of skewness for variables on the line. Our results are also illustrated via
numerical simulations.
