Benford                 package:VGAM                 R Documentation

_B_e_n_f_o_r_d'_s _D_i_s_t_r_i_b_u_t_i_o_n

_D_e_s_c_r_i_p_t_i_o_n:

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

_U_s_a_g_e:

     dbenf(x, ndigits=1, log=FALSE)
     pbenf(q, ndigits=1, log.p=FALSE)
     qbenf(p, ndigits=1)
     rbenf(n, ndigits=1)

_A_r_g_u_m_e_n_t_s:

    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)'.

_D_e_t_a_i_l_s:

     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.

_V_a_l_u_e:

     'dbenf' gives the density, 'pbenf' gives the distribution
     function, and 'qbenf' gives the quantile function, and 'rbenf'
     generates random deviates.

_A_u_t_h_o_r(_s):

     T. W. Yee

_R_e_f_e_r_e_n_c_e_s:

     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.

_E_x_a_m_p_l_e_s:

     x = c(0:10, NA, NaN, -Inf, Inf)
     dbenf(x)
     pbenf(x)

     ## Not run: 
     x = 1:9; # par(mfrow=c(2,1))
     barplot(dbenf(x), col = "lightblue", las=1, xlab="Leading digit",
             ylab = "Probability",
             main=paste("Benford's distribution",  sep=""),
             names.arg = as.character(x))

     hist(rbenf(n=1000), border = "blue", prob=TRUE,
          main="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(1:9, dbenf(1:9), col="red", type="h")
     points(1:9, dbenf(1:9), col="red")
     ## End(Not run)

