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This paper investigates the issue of cooperative spectrum sensing with a crowd of low-end personal spectrum sensors (such as smartphones, tablets, and in-vehicle sensors), where the sensing data from crowd sensors that may be unreliable, untrustworthy, or even malicious. Moreover, due to either unexpected equipment failures or malicious behaviors, every crowd sensor could sporadically and randomly contribute with abnormal data, which makes the existing cooperative sensing schemes ineffective. To tackle these challenges, we first propose a generalized modeling approach for sensing data with an arbitrary abnormal component. Under this model, we then analyze the impact of general abnormal data on the performance of the cooperative sensing, by deriving closed-form expressions of the probabilities of global false alarm and global detection. To improve sensing data quality and enhance cooperative sensing performance, we further formulate an optimization problem as stable principal component pursuit, and develop a data cleansing-based robust spectrum sensing algorithm to solve it, where the under-utilization of licensed spectrum bands and the sparsity of nonzero abnormal data are jointly exploited to robustly cleanse out the potential nonzero abnormal data component from the original corrupted sensing data. Extensive simulation results demonstrate that the proposed robust sensing scheme performs well under various abnormal data parameter configurations.