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We consider distributed estimation of a deterministic vector parameter from noisy sensor observations in a wireless sensor network (WSN). To meet stringent power and bandwidth budgets inherent in WSNs, local data dimensionality reduction is performed at each sensor to reduce the number of messages sent to a fusion center (FC). The problem of interest is to jointly design the compression matrices associated with those sensors, aiming at minimizing the estimation error at the FC. Such a dimensionality reduction problem is investigated in this paper. Specifically, we study an inhomogeneous environment where the noise covariance matrices across the sensors have the same correlation structure but with different scaling factors. Given a total number of messages sent to the FC, theoretical lower bounds on the estimation error of any compression strategy are derived. Compression strategies are developed to approach or even attain the corresponding theoretical lower bounds. Performance analysis and simulations are carried out to illustrate the optimality and effectiveness of the proposed compression strategies.