Benford's Distribution

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

Usage

dbenford(x, nDigits = 1, log = FALSE)

pbenford(q, nDigits = 1, lower.tail = TRUE, log.p = FALSE)

qbenford(p, nDigits = 1, lower.tail = TRUE, log.p = FALSE)

rbenford(n, nDigits = 1)

Arguments

x, q: a vector of quantiles, see nDigits.
nDigits: number of leading digits, either 1 or 2. If 1 then the support of the distribution is {1, ..., 9}, else {10, ..., 99}.
log: logical; if TRUE, densities are given as log(d).
lower.tail: logical; if TRUE (default), probabilities are $P(X \le x)$, otherwise $P(X > x)$.
log.p: logical; if TRUE, probabilities p are given as log(p).
p: a 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.

Value

dbenford() gives the density, pbenford() gives the distribution function, qbenford() gives the quantile function, and rbenford() generates random deviates.

Details

Benford's distribution (the significant-digit law) describes the probability distribution of the leading significant digit(s) in many naturally occurring numerical datasets. The probability of the first digit being d follows log10(1 + 1/d), with support on {1, ..., 9} for one leading digit or {10, ..., 99} for two leading digits. It is commonly applied in fraud detection and data quality assessment.

Benford's Law (aka the significant-digit law) is the empirical observation that in many naturally occurring 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. Newcomb. It started with 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 sources as different as atomic weights, baseball statistics, numerical data from Reader's Digest, and drainage areas of rivers.

Applications of Benford's Law have been as diverse as fraud detection in accounting and the design of computers.

Note

Based on code by T. W. Yee previously published in the VGAM package, adapted to conform to package standards.

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.

Examples

dbenford(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
pbenford(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

xx <- 1:9
barplot(dbenford(xx), col = "lightblue", las = 1, xlab = "Leading digit",
        ylab = "Probability", names.arg = as.character(xx),
        main = "Benford's distribution")


hist(rbenford(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, dbenford(xx), col = "red", type = "h")
points(xx, dbenford(xx), col = "red")