```help mata mm_benford()
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Title

mm_benford() -- Benford's Law (distribution of digits)

Syntax

real vector  mm_benford(digits [, position, base])

where

digits:  real vector providing the digits for which the probabilities
be computed; the digits must be in {0,...,base-1}

position:  real scalar specifying the position of the digits

base:  real scalar specifying the base of the numeric system

Description

mm_benford() returns the probabilities of the specified 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)

with d in {1,...,9}.

The general formula for the p-th digit in the base b number system is

b^(p-1) - 1
---
1      \      +      1   +
P(d, p, b) = -----         ln|1 + ----- |
ln(b)    /      +    k*b+d +
---
k = b^(p-2)

with d in {0,...,b-1} (see, e.g.,
http://www.mathpages.com/home/kmath302/kmath302.htm).

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).

Remarks

Examples

Fist digit Benford distribution:

: (1::9), mm_benford(1::9)
1             2
+-----------------------------+
1 |            1   .3010299957  |
2 |            2   .1760912591  |
3 |            3   .1249387366  |
4 |            4    .096910013  |
5 |            5    .079181246  |
6 |            6   .0669467896  |
7 |            7    .057991947  |
8 |            8   .0511525224  |
9 |            9   .0457574906  |
+-----------------------------+

Second digit Benford distribution:

: (0::9), mm_benford(0::9,2)
1             2
+-----------------------------+
1 |            0   .1196792686  |
2 |            1   .1138901034  |
3 |            2    .108821499  |
4 |            3   .1043295602  |
5 |            4   .1003082023  |
6 |            5   .0966772358  |
7 |            6   .0933747358  |
8 |            7   .0903519893  |
9 |            8   .0875700536  |
10 |            9   .0849973521  |
+-----------------------------+

Second digit Benford distribution in base 6:

: (0::5), mm_benford(0::5,2,6)
1             2
+-----------------------------+
1 |            0   .2019648465  |
2 |            1   .1841218513  |
3 |            2   .1697091954  |
4 |            3   .1577441976  |
5 |            4   .1476009617  |
6 |            5   .1388589474  |
+-----------------------------+

Conformability

mm_benford(digits, position, base):
digits:  r x 1 or 1 x c
position:  1 x 1
base:  1 x 1
result:  r x 1 or 1 x c.

Diagnostics

mm_benford(0,1,base) evaluates to missing.

Source code

mm_benford.mata

References

Benford, Frank (1938). The law of anomalous numbers. Proceedings of the
American Philosophical Society 78: 551–572.

Hill, Theodore P. (1998). The first digit phenomenon. American Scientist
86: 358.

Newcomb, Simon (1881). Note on the frequency of use of the different
digits in natural numbers. American Journal of Mathematics 4: 39–40.

Author

Ben Jann, ETH Zurich, jann@soz.gess.ethz.ch

Also see

Online:  help for [M-4] statistical, moremata

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