Experimental---This function returns previously numerical estimations of the L-moments of the Benford distribution (Benford's Law) given parameters defining the number of first M-significant digits and the numeric base.
For the first significant digits (\(d \in 1, \cdots, 9\)) (base 10) (designate as \(m = 1\)), the L-moments were estimated through very large sample-size simulation and sample L-moments computed ( lmoms), direct numerical integration (theoLmoms), and through numerical integration of the probability weighted moments and conversion to L-moments (pwm2lmom) as
$$\lambda_1 = 3.43908699617500524\mbox{,}$$
$$\lambda_2 = 1.34518434179517077\mbox{,}$$
$$\tau_3 = 0.24794090889493661\mbox{, and}$$
$$\tau_4 = 0.01614509742647182\mbox{.}$$
For the first two-significant digits (\(d \in 10, \cdots, 99\)) (base 10) (designate as \(m = 2\)), the L-moments were estimated through very large sample-size simulation, direct numerical integration (theoLmoms), and through numerical integration of the probability weighted moments and conversion to L-moments (pwm2lmom) as
$$\lambda_1 = 38.59062918136093145\mbox{,}$$
$$\lambda_2 = 13.81767809210059283\mbox{,}$$
$$\tau_3 = 0.22237541787527126\mbox{, and}$$
$$\tau_4 = 0.03541037418894027\mbox{.}$$
For the first three-significant digits (\(d \in 100, \cdots, 999\)) (base 10) (designate as \(m = 3\)), the L-moments were estimated through very large sample-size simulation, direct numerical integration (theoLmoms), and through numerical integration of the probability weighted moments and conversion to L-moments (pwm2lmom) as
$$\lambda_1 = 390.36783537821605705\mbox{,}$$
$$\lambda_2 = 138.21917489739223583\mbox{,}$$
$$\tau_3 = 0.22192482374529940\mbox{, and}$$
$$\tau_4 = 0.03571514686148788\mbox{.}$$
Source of the L-moments---The script inst/doc/benford/compLmomsBenford.R in the lmomco package sources is the authoritative source of the computation of the L-moments shown. Three methods are used, and the arithmetic average of the three provides the L-moments: (1) Probability-weighted simulation of the probability mass function (PMF) is used in very large sample size and sample L-moments computed by lmoms, (2) direct numerical integration for the theoretical L-moments of the quantile function (quaben) of the distribution that itself is from the cumulative distribution function (cdfben) that itself is from the PMF (pmfben), and (3) direct numerical integration of the probability-weighted moments of the quantile function (quaben) and subsequent linear system of equations to compute the L-moments. Each of the aforementioned methods result in numerical differences say at about the fourth decimal. (No previous description of the L-moments of the Benford distribution appear extant in the literature in July 2024.)
lmomben(para=list(para=c(1, 10)), ...)An R
list is returned.
Vector of the L-moments. First element is \(\lambda_1\), second element is \(\lambda_2\), and so on.
Vector of the L-moment ratios. Second element is \(\tau\), third element is \(\tau_3\) and so on.
Level of symmetrical trimming used in the computation, which is 0.
Level of left-tail trimming used in the computation, which is NULL.
Level of right-tail trimming used in the computation, which is NULL.
An attribute identifying the computational source of the L-moments: “lmomben”.
The number of first M-significant digits followed by the numerical base (only base10 supported) and the list structure mimics similar uses of the lmomco list structure. Default are the first significant digits and hence the digits 1 through 9.
Additional arguments to pass (not likely to be needed but changes in base handling might need this).
W.H. Asquith
cdfben, pmfben, quaben