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Anomaly detection incurs certain sampling and sensing costs and therefore it is of great importance to strike a balance between the detection accuracy and these costs. In this work, we study anomaly detection by considering the detection of threshold crossings in a stochastic time series without the knowledge of its statistics. To reduce the sampling cost in this detection process, we propose the use of hierarchical generative adversarial networks (GANs) to perform nonuniform sampling. In order to improve the detection accuracy and reduce the delay in detection, we introduce a buffer zone in the operation of the proposed GAN-based detector. In the experiments, we analyze the performance of the proposed hierarchical GAN detector considering the metrics of detection delay, miss rates, average cost of error, and sampling ratio. We identify the tradeoffs in the performance as the buffer zone sizes and the number of GAN levels in the hierarchy vary. We also compare the performance with that of a sampling policy that approximately minimizes the sum of average costs of sampling and error given the parameters of the stochastic process. We demonstrate that the proposed GAN-based detector can have significant performance improvements in terms of detection delay and average cost of error with a larger buffer zone but at the cost of increased sampling rates.