Benford's Distribution

Density, distribution function, quantile function, and random generation for Benford's distribution.

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 probabilities p are given as log(p).

Details

Benford's Law (aka the significant-digit law) is the empirical observation that in many naturally occuring tables of numerical data, the leading significant (nonzero) digit is not uniformly distributed in \(\{1,2,\ldots,9\}\). 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, 551–572.

Newcomb, S. (1881) Note on the Frequency of Use of the Different Digits in Natural Numbers. American Journal of Mathematics, 4, 39–40.

Author

T. W. Yee

Examples

dBenf(x <- c(0:10, NA, NaN, -Inf, Inf))
#>  [1] 0.00000000 0.30103000 0.17609126 0.12493874 0.09691001 0.07918125
#>  [7] 0.06694679 0.05799195 0.05115252 0.04575749 0.00000000         NA
#> [13]        NaN 0.00000000 0.00000000
pBenf(x)
#>  [1] 0.0000000 0.3010300 0.4771213 0.6020600 0.6989700 0.7781513 0.8450980
#>  [8] 0.9030900 0.9542425 1.0000000 1.0000000        NA       NaN 0.0000000
#> [15] 1.0000000

if (FALSE) { # \dontrun{
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")
} # }