Benford
Benford's Distribution
Density, distribution function, quantile function, and random generation for Benford's distribution.
 Keywords
 distribution
Usage
dBenf(x, ndigits = 1, log = FALSE)
pBenf(q, ndigits = 1, log.p = FALSE)
qBenf(p, ndigits = 1)
rBenf(n, ndigits = 1)
Arguments
 x, q

Vector of quantiles.
See
ndigits
.  p
 vector of probabilities.
 n
 number of observations. A single positive integer.
Else if
length(n) > 1
then the length is taken to be the number required.  ndigits

Number of leading digits, either 1 or 2.
If 1 then the support of the distribution is {1,...,9}, else
{10,...,99}.
 log, log.p

Logical.
If
log.p = TRUE
then all probabilitiesp
are given aslog(p)
.
Details
Benford's Law (aka the significantdigit law) is the empirical observation that in many naturally occuring tables of numerical data, the leading significant (nonzero) digit is not uniformly distributed in $1:9$. Instead, the leading significant digit ($=D$, say) obeys the law $$P(D=d) = \log_{10} \left( 1 + \frac1d \right)$$ for $d=1,\ldots,9$. This means the probability the first significant digit is 1 is approximately $0.301$, etc.
Benford's Law was apparently first discovered in 1881 by astronomer/mathematician S. Newcombe. It started by the observation that the pages of a book of logarithms were dirtiest at the beginning and progressively cleaner throughout. In 1938, a General Electric physicist called F. Benford rediscovered the law on this same observation. Over several years he collected data from different sources as different as atomic weights, baseball statistics, numerical data from Reader's Digest, and drainage areas of rivers.
Applications of Benford's Law has been as diverse as to the area of fraud detection in accounting and the design computers.
Value
dBenf
gives the density,
pBenf
gives the distribution function, and
qBenf
gives the quantile function, and
rBenf
generates random deviates.Source
These functions were previously published as dbenf()
etc. in the VGAM package and have been
integrated here without logical changes.
References
Benford, F. (1938) The Law of Anomalous Numbers. Proceedings of the American Philosophical Society, 78, 551572.
Newcomb, S. (1881) Note on the Frequency of Use of the Different Digits in Natural Numbers. American Journal of Mathematics, 4, 3940.
Examples
dBenf(x < c(0:10, NA, NaN, Inf, Inf))
pBenf(x)
## Not run:
# xx < 1:9
# barplot(dBenf(xx), col = "lightblue", las = 1, xlab = "Leading digit",
# ylab = "Probability", names.arg = as.character(xx),
# main = paste("Benford's distribution", sep = ""))
#
# hist(rBenf(n = 1000), border = "blue", prob = TRUE,
# main = "1000 random variates from Benford's distribution",
# xlab = "Leading digit", sub="Red is the true probability",
# breaks = 0:9 + 0.5, ylim = c(0, 0.35), xlim = c(0, 10.0))
# lines(xx, dBenf(xx), col = "red", type = "h")
# points(xx, dBenf(xx), col = "red")
# ## End(Not run)