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The climate system occasionally experiences an abrupt change. However, it is very difficult to predict this change at present. Fortunately, some generic properties have been revealed before several different types of dynamical system near their own critical thresholds. These properties provide a possible way to give an early warning for an impending abrupt climate change. Therefore, it is important to evaluate the applicability of an early warning indicator of an abrupt change. On the basis of several simple fold models, we have systematically investigated the performance of the kurtosis coefficient as an early warning signal for an upcoming abrupt climate change. The testing results indicate that the kurtosis coefficient is a reliable warning indicator in most of cases whether a critical control parameter or the strength of an external forcing approaches a critical point. However, the strong noise can greatly shorten the effective warning time, and also can result in the reduction of the magnitude of a kurtosis coefficient when a dynamical system approaches its critical threshold. The missing data has almost no effect on the kurtosis coefficient in all of tests, even it is true when the missing data accounts for 20% of the total sample. We also found that the kurtosis coefficient does not work in some cases, which means that the kurtosis coefficient is not a universal early warning signal for an upcoming abrupt change.