In this paper, a distributed method for fault detection using sensor networks is proposed. Each sensor communicates only with neighboring nodes to compute locally an estimate of the state of the system to monitor. A residual is defined and suitable stochastic thresholds are designed, allowing to set the parameters so to guarantee a maximum false alarms probability. The main novelty and challenge of the proposed approach consists in addressing the individual correlations between the state, the measurements, and the noise components, thus significantly generalising the estimation methodology compared to previous results. No assumptions on the probability distribution family are needed for the noise variables. Simulation results show the effectiveness of the proposed method, including an extensive sensitivity analysis with respect to fault magnitude and measurement noise.