Benford's Law - Statistical Tests

Statistical Tests

Statistical tests examining the fit of Benford's law to data have more power when the data values span several orders of magnitude. Since many data samples typically do not have this range, numerical transformation of the data to a base other than 10 may be useful before testing.

Although the chi squared test has been used to test for compliance with Benford's law it has low statistical power when used with small samples.

The Kolmogorov-Smirnov test and the Kuiper test are more powerful when the sample size is small particularly when Stephen's corrective factor is used. These tests may be overly conservative when applied to discrete distribution. Values for the Benford test have been generated by Morrow(2010), and the critical values of the test statistics are shown below:

Two alternative tests specific to this law have been published: first, the max (m) statistic from Leemis et al (2000) is given by

and secondly, the distance (d) statistic from Cho and Gaines is given by

where FSD is the First Significant Digit. Morris has determined the critical values for both these statistics, which are shown below:

Nigrini has suggested the use of a z statistic

with

where || is the absolute value, n is the sample size, 1/(2n) is a continuity correction factor, p_e is the proportion expected from Benford's law and p_o is the observed proportion in the sample.

Morris has also shown that for any random variable X (with a continuous pdf) divided by its standard deviation (σ), a value A can be found such that the probability of the distribution of the first significant digit of the random variable ( X / σ )A will differ from Benford's Law by less than ε > 0. The value of A depends on the value of ε and the distribution of the random variable.

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