This paper studies minimal-energy decentralized estimation in sensor networks under best-linear-unbiased-estimator fusion rule. While most of the existing related works require the knowledge of instantaneous noise variances for energy allocation, the proposed approach instead relies on an associated statistical model. The minimization of total energy is subject to certain performance constraint in terms of mean square error (MSE) averaged over the noise variance distribution. A closed-form formula for the overall MSE metric is derived, based on which the problem can be reformulated in the form of convex optimization and is shown to yield an analytic solution. The proposed method shares several attractive features of the existing designs via instantaneous noise variances; through simulations it is seen to significantly improve the energy efficiency against the uniform allocation scheme.