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We derive linear estimators of stationary random signals based on reduced-dimensionality observations collected at distributed sensors and communicated to a fusion center over wireless links. Dimensionality reduction compresses sensor data to meet low-power and bandwidth constraints, while linearity in compression and estimation are well motivated by the limited computing capabilities wireless sensor networks are envisioned to operate with, and by the desire to estimate random signals from observations with unknown probability density functions. In the absence of fading and fusion center noise (ideal links), we cast this intertwined compression-estimation problem in a canonical correlation analysis framework and derive closed-form mean-square error (MSE) optimal estimators along with coordinate descent suboptimal alternatives that guarantee convergence at least to a stationary point. Likewise, we develop estimators based on reduced-dimensionality sensor observations in the presence of fading and additive noise at the fusion center (nonideal links). Performance analysis and corroborating simulations demonstrate the merits of the novel distributed estimators relative to existing alternatives.