In this paper, we consider the problem of parameter estimation over sensor networks under data rate constraint. A general additive quantization model is introduced to capture the data rate constraint. Existing works on the effect of the additive model on standard consensus algorithms show that convergence can be guaranteed only if the quantization error variances form a convergent series. We propose to incorporate a moving average step into the consensus algorithm to smear out the randomness caused by quantization errors. It is shown that the proposed algorithm achieves the performance of the optimal centralized sample mean estimator even if the quantization error variances are not vanishing. This is guaranteed by establishing a law of the iterated logarithm for weighted sums of independent random vectors. Moreover, an explicit bound of the rate of convergence is given to quantify its almost sure performance. Finally, simulations are provided to validate the theoretical results.