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In this paper, tight upper and lower bounds are derived on the weighted sumof minimum mean-squared errors for additive Gaussian noise channels. The boundsare obtained by constraining the input distribution to be close to a Gaussianreference distribution in terms of the Kullback--Leibler divergence. Thedistributions that attain these bounds are shown to be Gaussian whosecovariance matrices are defined implicitly via systems of matrix equations.Furthermore, the estimators that attain the upper bound are shown to be minimaxrobust against deviations from the assumed input distribution. The lower boundprovides a potentially tighter alternative to well-known inequalities such asthe Cram\'{e}r--Rao lower bound. Numerical examples are provided to verify thetheoretical findings of the paper. The results derived in this paper can beused to obtain performance bounds, robustness guarantees, and engineeringguidelines for the design of local estimators for distributed estimationproblems which commonly arise in wireless communication systems and sensornetworks.

Tight Bounds on the Weighted Sum of MMSEs with Applications in Distributed Estimation