Video anomaly detection is to determine whether there are any abnormal events, behaviors or objects in a given video, which enables effective and intelligent public safety management. As video anomaly labeling is both time-consuming and expensive, most existing works employ unsupervised or weakly supervised learning methods. This paper focuses on weakly supervised video anomaly detection, in which the training videos are labeled whether or not they contain any anomalies, but there is no information about which frames the anomalies are located. However, the uncertainty of weakly labeled data and the large model size prevent existing methods from wide deployment in real scenarios, especially the resource-limit situations such as edge-computing. In this paper, we develop a lightweight video anomaly detection model. On the one hand, we propose an adaptive instance selection strategy, which is based on the model's current status to select confident instances, thereby mitigating the uncertainty of weakly labeled data and subsequently promoting the model's performance. On the other hand, we design a lightweight multi-level temporal correlation attention module and an hourglass-shaped fully connected layer to construct the model, which can reduce the model parameters to only 0.56\% of the existing methods (e.g. RTFM). Our extensive experiments on two public datasets UCF-Crime and ShanghaiTech show that our model can achieve comparable or even superior AUC score compared to the state-of-the-art methods, with a significantly reduced number of model parameters.