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Benf

Benford's Distribution


Description

Density, distribution function, quantile function, and random generation for Benford's 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 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 1:9. Instead, the leading significant digit (=D, say) obeys the law

P(D=d) = log10(1 + 1/d)

for d=1,…,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.

Author(s)

T. W. Yee

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.

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)

DescTools

Tools for Descriptive Statistics

v0.99.41
GPL (>= 2)
Authors
Andri Signorell [aut, cre], Ken Aho [ctb], Andreas Alfons [ctb], Nanina Anderegg [ctb], Tomas Aragon [ctb], Chandima Arachchige [ctb], Antti Arppe [ctb], Adrian Baddeley [ctb], Kamil Barton [ctb], Ben Bolker [ctb], Hans W. Borchers [ctb], Frederico Caeiro [ctb], Stephane Champely [ctb], Daniel Chessel [ctb], Leanne Chhay [ctb], Nicholas Cooper [ctb], Clint Cummins [ctb], Michael Dewey [ctb], Harold C. Doran [ctb], Stephane Dray [ctb], Charles Dupont [ctb], Dirk Eddelbuettel [ctb], Claus Ekstrom [ctb], Martin Elff [ctb], Jeff Enos [ctb], Richard W. Farebrother [ctb], John Fox [ctb], Romain Francois [ctb], Michael Friendly [ctb], Tal Galili [ctb], Matthias Gamer [ctb], Joseph L. Gastwirth [ctb], Vilmantas Gegzna [ctb], Yulia R. Gel [ctb], Sereina Graber [ctb], Juergen Gross [ctb], Gabor Grothendieck [ctb], Frank E. Harrell Jr [ctb], Richard Heiberger [ctb], Michael Hoehle [ctb], Christian W. Hoffmann [ctb], Soeren Hojsgaard [ctb], Torsten Hothorn [ctb], Markus Huerzeler [ctb], Wallace W. Hui [ctb], Pete Hurd [ctb], Rob J. Hyndman [ctb], Christopher Jackson [ctb], Matthias Kohl [ctb], Mikko Korpela [ctb], Max Kuhn [ctb], Detlew Labes [ctb], Friederich Leisch [ctb], Jim Lemon [ctb], Dong Li [ctb], Martin Maechler [ctb], Arni Magnusson [ctb], Ben Mainwaring [ctb], Daniel Malter [ctb], George Marsaglia [ctb], John Marsaglia [ctb], Alina Matei [ctb], David Meyer [ctb], Weiwen Miao [ctb], Giovanni Millo [ctb], Yongyi Min [ctb], David Mitchell [ctb], Franziska Mueller [ctb], Markus Naepflin [ctb], Daniel Navarro [ctb], Henric Nilsson [ctb], Klaus Nordhausen [ctb], Derek Ogle [ctb], Hong Ooi [ctb], Nick Parsons [ctb], Sandrine Pavoine [ctb], Tony Plate [ctb], Luke Prendergast [ctb], Roland Rapold [ctb], William Revelle [ctb], Tyler Rinker [ctb], Brian D. Ripley [ctb], Caroline Rodriguez [ctb], Nathan Russell [ctb], Nick Sabbe [ctb], Ralph Scherer [ctb], Venkatraman E. Seshan [ctb], Michael Smithson [ctb], Greg Snow [ctb], Karline Soetaert [ctb], Werner A. Stahel [ctb], Alec Stephenson [ctb], Mark Stevenson [ctb], Ralf Stubner [ctb], Matthias Templ [ctb], Duncan Temple Lang [ctb], Terry Therneau [ctb], Yves Tille [ctb], Luis Torgo [ctb], Adrian Trapletti [ctb], Joshua Ulrich [ctb], Kevin Ushey [ctb], Jeremy VanDerWal [ctb], Bill Venables [ctb], John Verzani [ctb], Pablo J. Villacorta Iglesias [ctb], Gregory R. Warnes [ctb], Stefan Wellek [ctb], Hadley Wickham [ctb], Rand R. Wilcox [ctb], Peter Wolf [ctb], Daniel Wollschlaeger [ctb], Joseph Wood [ctb], Ying Wu [ctb], Thomas Yee [ctb], Achim Zeileis [ctb]
Initial release
2021-04-09

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