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Consider a wireless sensor network (WSN) with a fusion center (FC) deployed to estimate signal parameters from noisy sensor measurements. If the WSN has a large number of low-cost, battery-operated sensor nodes with limited transmission bandwidth, then conservation of transmission resources (power and bandwidth) is paramount. To this end, the present paper develops a novel data reduction method which requires no inter-sensor collaboration and results in only a subset of the sensor measurements transmitted to the FC. Using interval censoring as a data-reduction method, each sensor decides separately whether to censor its acquired measurements based on a rule that promotes censoring of measurements with least impact on the estimator mean-square error (MSE). Leveraging the statistical distribution of sensor data, the censoring mechanism and the received uncensored data, FC-based estimators are derived for both deterministic (via maximum likelihood estimation) and random parameters (via maximum a posteriori probability estimation) for a linear-Gaussian model. Quantization of the uncensored measurements at the sensor nodes offers an additional degree of freedom in the resource conservation versus estimator MSE reduction tradeoff. Cramér-Rao bound analysis for the different censor-estimators and censor-quantizer estimators is also provided to benchmark and facilitate MSE-based performance comparisons. Numerical simulations corroborate the analytical findings and demonstrate that the proposed censoring-estimation approach performs competitively with alternative methods, under different sensing conditions, while having lower computational complexity.