{smcl}
{* 12dec2007}{...}
{cmd:help mata mm_benford()}
{hline}
{title:Title}
{p 4 14 2}
{bf:mm_benford() -- Benford's Law (distribution of digits)}
{title:Syntax}
{p 8 23 2}
{it:real vector}{space 1}
{cmd:mm_benford(}{it:digits} [{cmd:,} {it:position}{cmd:,} {it:base}]{cmd:)}
{pstd}
where
{p 7 16 2}
{it:digits}: {it:real vector} providing the digits for which the probabilities
be computed; the digits must be in {c -(}0,...,{it:base}-1{c )-}
{p 5 16 2}
{it:position}: {it:real scalar} specifying the position of the digits
{p 9 16 2}
{it:base}: {it:real scalar} specifying the base of the numeric system
{title:Description}
{pstd}
{cmd:mm_benford()} returns the probabilities of the specified {it:digits} according
to Benford's Law (Newcomb 1881, Benford 1938, Hill 1998). The formula for the
1st (leading) digit in the base 10 number system is:
P(d) = log10(1 + 1/d)
{pstd}
with d in {c -(}1,...,9{c )-}.
{pstd}The general formula for the p-th digit in the base b number system is
b^(p-1) - 1
{hline 3}
1 \ {c TLC} 1 {c TRC}
P(d, p, b) = {hline 5} ln{c |}1 + {hline 5} {c |}
ln(b) / {c BLC} k*b+d {c BRC}
{hline 3}
k = b^(p-2)
{pstd}
with d in {c -(}0,...,b-1{c )-} (see, e.g.,
{browse "http://www.mathpages.com/home/kmath302/kmath302.htm"}).
{pstd}The limit of
P(d) with rising p is 1/b. Computation becomes exceedingly
intensive with rising p since the number of operations is in the order
of b^p - b^(p-1).
{title:Remarks}
{pstd} Examples
{pstd} Fist digit Benford distribution:
{com}: (1::9), mm_benford(1::9)
{res} {txt} 1 2
{c TLC}{hline 29}{c TRC}
1 {c |} {res} 1 .3010299957{txt} {c |}
2 {c |} {res} 2 .1760912591{txt} {c |}
3 {c |} {res} 3 .1249387366{txt} {c |}
4 {c |} {res} 4 .096910013{txt} {c |}
5 {c |} {res} 5 .079181246{txt} {c |}
6 {c |} {res} 6 .0669467896{txt} {c |}
7 {c |} {res} 7 .057991947{txt} {c |}
8 {c |} {res} 8 .0511525224{txt} {c |}
9 {c |} {res} 9 .0457574906{txt} {c |}
{c BLC}{hline 29}{c BRC}{txt}
{pstd} Second digit Benford distribution:
{com}: (0::9), mm_benford(0::9,2)
{res} {txt} 1 2
{c TLC}{hline 29}{c TRC}
1 {c |} {res} 0 .1196792686{txt} {c |}
2 {c |} {res} 1 .1138901034{txt} {c |}
3 {c |} {res} 2 .108821499{txt} {c |}
4 {c |} {res} 3 .1043295602{txt} {c |}
5 {c |} {res} 4 .1003082023{txt} {c |}
6 {c |} {res} 5 .0966772358{txt} {c |}
7 {c |} {res} 6 .0933747358{txt} {c |}
8 {c |} {res} 7 .0903519893{txt} {c |}
9 {c |} {res} 8 .0875700536{txt} {c |}
10 {c |} {res} 9 .0849973521{txt} {c |}
{c BLC}{hline 29}{c BRC}{txt}
{pstd} Second digit Benford distribution in base 6:
{com}: (0::5), mm_benford(0::5,2,6)
{res} {txt} 1 2
{c TLC}{hline 29}{c TRC}
1 {c |} {res} 0 .2019648465{txt} {c |}
2 {c |} {res} 1 .1841218513{txt} {c |}
3 {c |} {res} 2 .1697091954{txt} {c |}
4 {c |} {res} 3 .1577441976{txt} {c |}
5 {c |} {res} 4 .1476009617{txt} {c |}
6 {c |} {res} 5 .1388589474{txt} {c |}
{c BLC}{hline 29}{c BRC}{txt}
{title:Conformability}
{cmd:mm_benford(}{it:digits}{cmd:,} {it:position}{cmd:,} {it:base}{cmd:)}:
{it:digits}: {it:r x} 1 or 1 {it:x c}
{it:position}: 1 {it:x} 1
{it:base}: 1 {it:x} 1
{it:result}: {it:r x} 1 or 1 {it:x c}.
{title:Diagnostics}
{p 4 4 2}{cmd:mm_benford(0,1,}{it:base}{cmd:)} evaluates to missing.
{title:Source code}
{p 4 4 2}
{help moremata_source##mm_benford:mm_benford.mata}
{title:References}
{phang}
Benford, Frank (1938). The law of anomalous numbers. Proceedings of the
American Philosophical Society 78: 551-572.
{phang}
Hill, Theodore P. (1998). The first digit phenomenon. American Scientist 86: 358.
{phang}
Newcomb, Simon (1881). Note on the frequency of use of the different
digits in natural numbers. American Journal of Mathematics 4: 39-40.
{p_end}
{title:Author}
{p 4 4 2} Ben Jann, University of Bern, jann@soz.unibe.ch
{title:Also see}
{p 4 13 2}
Online: help for
{bf:{help m4_statistical:[M-4] statistical}},
{bf:{help moremata}}
{p_end}