### Abstract

Upper and lower bounds on the minimum mean square error for additive noise channels are derived when the input distribution is constrained to be close to a Gaussian reference distribution in terms of the Kullback-Leibler divergence. The upper bound is tight and is attained by a Gaussian distribution whose mean is identical to that of the reference distribution and whose covariance matrix is defined implicitly via a system of non-linear equations. The estimator that attains the upper bound is identified as a minimax optimal estimator that is robust against deviations from the assumed prior. The lower bound provides an alternative to well-known inequalities in estimation and information theory-such as the Cramér-Rao lower bound, Stam's inequality, or the entropy power inequality-that is potentially tighter and defined for a larger class of input distributions. Several examples of applications in signal processing and information theory illustrate the usefulness of the proposed bounds in practice.

Original language | English (US) |
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Article number | 8890879 |

Pages (from-to) | 6352-6367 |

Number of pages | 16 |

Journal | IEEE Transactions on Signal Processing |

Volume | 67 |

Issue number | 24 |

DOIs | |

State | Published - Dec 15 2019 |

### All Science Journal Classification (ASJC) codes

- Signal Processing
- Electrical and Electronic Engineering

### Keywords

- Cramér-Rao bound
- MMSE bounds
- Stam's inequality
- entropy power inequality
- minimax optimization
- robust estimation

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## Cite this

*IEEE Transactions on Signal Processing*,

*67*(24), 6352-6367. [8890879]. https://doi.org/10.1109/TSP.2019.2951221