We study the problem of minimal-energy decentralized estimation via sensor networks with the best-linear-unbiased-estimator fusion rule. While most of the existing solutions 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 a performance constraint in terms of the reciprocal of mean square errors averaged over the considered distribution. A closed-form formula for such a mean distortion metric, as well as an associated tractable lower bound, is derived. By imposing a target distortion constraint in terms of this bound and further through feasible set relaxation, the problem can be reformulated in the form of convex optimization and is then analytically solved. 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.