In wireless sensor networks, many monitoring problems can be cast in the form of distributed estimation. If the data links from the sensor nodes to the fusion center have limited capacity, there is a tradeoff between estimation precision and transmission rate. This kind of decentralized estimation system is equivalent to the so-called indirect multiterminal source coding problem, and the Berger-Tung inner bound is the best known achievable rate region boundary. In this paper, we attempt to evaluate the Berger-Tung sum rate for a vector source with correlated components. First we formulate the sum rate as a nonconvex optimization problem with a distortion constraint. Then we propose a method to find an approximate solution. Numerical experiments show the approximate solution is accurate if the required distortion level is relatively small. Its appropriateness is also verified by simulation results based on practical quantizer design.