Tests of Benford's Law With Common Distributions
Benford's law was empirically tested against the numbers (up to the 10th digit) generated by a number of important distributions. including the uniform distribution, the exponential distribution, the half-normal distribution, the right-truncated normal, the normal distribution, the chi square distribution and the log normal distribution In addition to these the ratio distribution of two uniform distributions, the ratio distribution of two exponential distributions, the ratio distribution of two half-normal distributions, the ratio distribution of two right-truncated normal distributions, the ratio distribution of two chi-square distributions (the F distribution) and the log normal distribution were tested.
The uniform distribution as might be expected does not obey Benford's law. In contrast, the ratio distribution of two uniform distributions is well described by Benford's law. Benford's law also describes the exponential distribution and the ratio distribution of two exponential distributions well. Although the half-normal distribution does not obey Benford's law, the ratio distribution of two half-normal distributions does. Neither the right-truncated normal distribution nor the ratio distribution of two right-truncated normal distributions are well described by Benford's law. This is not surprising as this distribution is weighted towards larger numbers. Neither the normal distribution nor the ratio distribution of two normal distributions (the Cauchy distribution) obey Benford's law. The fit of chi square distribution depends on the degrees of freedom (df) with good agreement with df = 1 and decreasing agreement as the df increases. The F distribution is fitted well for low degrees of freedom. With increasing dfs the fit decreases but much more slowly than the chi square distribution. The fit of the log-normal distribution depends on the mean and the variance of the distribution. The variance has a much greater effect on the fit than does the mean. Larger values of both parameters result in better agreement with the law. The ratio of two log normal distributions is a log normal so this distribution was not examined.
Other distributions that have been examined include the Muth distribution, Gompertz distribution, Weibull distribution, gamma distribution, log-logistic distribution and the exponential power distribution all of which show reasonable agreement with the law. The Gumbel distribution - a density increases with increasing value of the random variable - does not show agreement with this law.
Read more about this topic: Benford's Law
Famous quotes containing the words tests of, tests, law and/or common:
“Every perversion has survived many tests of its capabilities.”
—Mason Cooley (b. 1927)
“Letters have to pass two tests before they can be classed as good: they must express the personality both of the writer and of the recipient.”
—E.M. (Edward Morgan)
“Half the testimony in the Bobbitt case sounded like Sally Jesse Raphael. Juries watch programs like this and are ready to listen.”
—William Geimer, U.S. law educator. New York Times, p. B18 (January 28, 1994)
“Nature seems to have taken a particular Care to disseminate her Blessings among the different Regions of the World, with an Eye to this mutual Intercourse and Traffick among Mankind, that the Natives of the several Parts of the Globe might have a kind of Dependance [sic] upon one another, and be united together by their common Interest.”
—Joseph Addison (1672–1719)