To guarantee its integrity, a wireless sensor network needs to efficiently detect faulty nodes producing erroneous measurements. This paper proposes a fully distributed fault detection algorithm. A node first collects the measurements of its neighborhood, processes them to decide whether they contain outliers, and broadcasts the result. Then, it decides autonomously about its functioning status. The detection algorithm is proposed in two variants, depending on the proportion of faulty nodes in the network. A theoretical analysis of the probability of error and of the convergence of the algorithm is provided. The tradeoff between false alarm probability and detection probability is characterized using simulation.