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When Does the Second-Digit Benford’s Law-Test Signal an Election Fraud?

Summary Detecting election fraud with a simple statistical method and minimal information makes the application of Benford’s Law quite promising for a wide range of researchers. Whilst its specific form, the Second-Digit Benford’s Law (2BL)-test, is increasingly applied to fraud suspected elections, concerns about the validity of its test results have been raised. One important caveat of this kind of research is that the 2BL-test has been applied mostly to fraud suspected elections. Therefore, this article will apply the test to the 2009 German Federal Parliamentary Election against which no serious allegation of fraud has been raised. Surprisingly, the test results indicate that there should be electoral fraud in a number of constituencies. These counter intuitive results might be due to the naive application of the 2BL-test which is based on the conventional χ2 distribution. If we use an alternative distribution based on simulated election data, the 2BL-test indicates no significant deviation. Using the simulated election data, we also identified under which circumstances the naive application of the 2BL-test is inappropriate. Accordingly, constituencies with homogeneous precincts and a specific range of vote counts tend to have a higher value for the 2BL statistic.