From the power law to extreme value mixture distributions
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The power law is useful in describing count phenomena such as network degrees and word frequencies. With a single parameter, it captures the main feature that the frequencies are linear on the log-log scale. Nevertheless, there have been criticisms of the power law, and various approaches have been proposed to resolve issues such as selecting the required threshold and quantifying the uncertainty around it, and to test hypotheses on whether the data could have come from the power law. As extreme value theory generalises the (continuous) power law, it is natural to consider the former as a solution to these problems around the latter. In this paper, we propose two extreme value mixture distributions, in one of which the power law is incorporated, without the need of pre-specifying the threshold. The proposed distributions are shown to fit the data well, quantify the threshold uncertainty in a natural way, and satisfactorily answer whether the power law is useful enough.
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