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Abstract

A simple one-parameter analytical extension of Benford’s law for first digits of numerical data is constructed. Based on the maximum likelihood method, the fitting capabilities of the new distribution is illustrated at some interesting and important integer sequences including the numeri ideoni, the Keith, Princeton, lucky, Ulam and Bell numbers, as well as the sequence of primes. Benford’s law of the mixing of the considered data sets is rejected at the 5% significance level while the generalized Benford law is accepted with a 25% p-value. Confirming the statistical evidence, it is shown that the first digits of the Bell numbers satisfy Benford’s law.
A GENERALIZED BENFORD LAW AND ITS APPLICATION
By Werner Hürlimann, Switzerland
Abstract.
A simple one-parameter analytical extension of Benford’s law for first digits of numerical
data is constructed. Based on the maximum likelihood method, the fitting capabilities of the
new distribution is illustrated at some interesting and important integer sequences including
the numeri ideoni, the Keith, Princeton, lucky, Ulam and Bell numbers, as well as the
sequence of primes. Benford’s law of the mixing of the considered data sets is rejected at the
5% significance level while the generalized Benford law is accepted with a 25% p-value.
Confirming the statistical evidence, it is shown that the first digits of the Bell numbers satisfy
Benford’s law.
Keywords : first digit, Benford’s law, uniform distribution, triangular distribution,
two-sided power distribution, integer sequences, mixing sequences, Bell number
MSC 2000 : 11B73, 11B83, 11K31, 62E15, 62E17
1. Introduction.
Since Newcomb(1881) and Benford(1938) it is known that many numerical data sets obey
Benford’s law. To be specific, if the random variable X, which describes the first significant
digit in a numerical table, is Benford distributed, then
()
()
PX d d== +
log 1
1
,
{}
d 19,..., . (1.1)
Mathematical explanations of this law have been proposed by Pinkham(1961),
Cohen(1976), Hill(1995a/b/c,97,98), Allart(1997), Janvresse and de la Rue(2003). In recent
years an upsurge of applications of Benford’s law have appeared including work by
Becker(1982), Burke and Kincanon(1991), Buck et al.(1993), Ley(1996), Nigrini(1996/99),
Nigrini and Mittermaier(1997), Vogt(2000), Tolle et al.(2000) and Berger et al.(2002).
Hill(1995c) also suggested to switch the attention to probability distributions that follow
or closely approximate Benford’s law. Papers along this path include Leemis et al.(2000) and
Engel and Leuenberger(2003). Some survival distributions, which satisfy exactly Benford’s
law, are known. However, there does not seem to exist simple analytical distributions, which
include as special case Benford’s law. Combining facts from Leemis et al.(2000) and Dorp
and Kotz(2002), such a simple one-parameter family of distributions is constructed.
The interest of the enlarged Benford law is two-fold. First, it may provide a better fit of
the data than Benford’s law. Second, it yields a simple statistical procedure to validate
Benford’s law. If Benford’s model is sufficiently “close” to the one-parameter extended
model, then it will be retained. These points will be illustrated by some typical examples.
2
2. A generalized Benford distribution.
If T denotes a random lifetime with survival distribution
S
t
P
T
t
() ( )=≥, then the value
Y of the first significant digit in the lifetime T has probability distribution
() ( )
{}
{}
PY y S y S y y
ii
i
== +
=−
( )() , ,...,10 1 10 1 9 . (2.1)
Alternatively, if D denotes the integer-valued random variable satisfying
10 10
1DD
T≤<
+
, (2.2)
then the first significant digit can be written in terms of T and D as
[][]
YT
DTD
=⋅ =
−−
10 10
log
, (2.3)
where
[]
x denotes the greatest integer less than or equal to x. In particular, if the random
variable
ZTD=−log is uniformly distributed as U (,)0 1 , then the first significant digit Y
is exactly Benford distributed. Starting from the uniform random variable
W
U= (,)02 or the
triangular random variable W Triangular= (,,)0 1 2 with probability density function
fw w
W
()= if
()
w 01, and fw w
W
()=−2 if
[
)
w 12, , one shows that the random
lifetime T
W
=10 generates the first digit Benford distribution (Leemis et al.(2000),
Examples 1 and 2).
A simple parametric distribution, which includes as special cases both the above uniform
and triangular distributions, is the two-sided power random variable
()
WTSP c=
α
, recently
considered in Dorp and Kotz(2002) with probability density function
fw
W
()=
()
()
cw
c
cw
c
w
w
2
1
2
2
2
1
0
2
α
α
α
α
<≤
≤<
,,
,.
(2.4)
If c
=1 then
W
U= (,)02 , and if c ==21,
α
then W Triangular= (,,)012 . This
observation shows that the random lifetime
()
T
TSP c
=10
1,
will generate first digit
distributions closely related to Benfords distribution, at least if c is close to 1 or 2.
Theorem 2.1. Let
()
WTSPc= 1, be the two-sided power random variable with probability
density function
fw
W
()=
()
c
c
c
c
ww
ww
2
1
2
1
01
212
<≤
−≤<
,,
,,
(2.5)
and let the integer-valued random variable D satisfy D
W
D≤<+1. Then the first digit
random variable
[]
Y
WD
=
10 has the one-parameter generalized Benford probability density
function
[][][ ][]
{}
{}
fy y y y y y
Y
cc c c
( ) log( ) log log( ) log , ,..., .=+++
1
2
111119 (2.6)
3
Proof. The probability density function of T
W
=10 is given by
ft
T
()=
()
()
c
t
t
c
c
t
t
c
t
t
210 10
1
210 10
1
010
2 10 100
<≤
−≤<
ln
ln
ln
ln
ln
ln
,,
,.
It follows that the first significant digit of T, namely
[]
YT
D
=⋅
10 , has probability density
function
fy ftdt ftdt
YT
y
y
T
y
y
() () ()
()
=+
++
∫∫
1
10
10 1
.
Making the change of variable
ut= ln / ln10 , one obtains
f y cu du c u du
Y
c
y
y
c
y
y
() ( )
log
log( )
log
log( )
=+
+
+
++
∫∫
1
2
2
1
1
1
1
11
,
which upon integration immediately implies (2.6).
3. Fitting the first digit distributions of some integer sequences.
Maximum likelihood estimation of the generalized Benford distribution is straightforward.
To simplify notation, set
{}
ayb yy
Yy
==+log , log( ), ,...,1 1 9 , and consider the
derivative of Lfy
yY
= ln ( ) with respect to the parameter c, which is given by
c
L
y
=
ln ln( ) ( )
()
,,
ln ln ln( ) ( ) ln( ) ( )
()
, ,..., ,
ln ln()()
()
,.
bb b b
bb
y
bb a a b b a a
bb
y
aa a a
aa
y
cc
cc
yy
c
YY
c
YY
c
YY
c
cc
cc
cc
11 1 1
11
11
99 9 9
99
11
11
1
11 11
11
28
11
11
9
⋅−
−− +
=
⋅− +
−− +
=
−⋅+
−+
=
If
{}
Ny
y
, ,..., 1 9 , is the number of observations with first digit y, then the maximum
likelihood estimator (MLE) of c will solve the non-linear equation
N
c
L
yy
y
⋅=
=
1
9
0
. (3.1)
In some cases, more than one solution may exist, usually near c = 1 and c = 2. In this
situation, the solution with greatest value of the log-likelihood
ln ln ( )LNfy
yY
y
=⋅
=
1
9
will
be chosen as MLE.
4
The fitting capabilities of the new distribution is illustrated at some interesting and
important integer sequences. The first digit occurrences of the analyzed sequences are listed
in Table 3.1. Then, we give some comments on the interest of these sequences. Statistical
results are summarized in Table 3.2. Besides the value of the MLE of the parameter c, we list
the negative log-likelihood, the chi-square values and their corresponding p-values. Finally,
the obtained results are discussed.
Table 3.1 : First digit distributions of some integer sequences
First digit occurrences
Name of sequence Sample
size
1 2 3 4 5 6 7 8 9
Numeri ideoni 65 20 12 9 7 4 2 5 4 2
Keith number 71 23 10 10 5 3 5 9 2 4
Princeton number 25 7 2 3 3 2 3 2 1 2
Lucky number 45 19 8 4 2 1 3 4 1 3
Ulam number 44 20 6 3 3 2 3 2 3 2
Fibonacci number 100 30 18 13 9 8 6 5 7 4
Bell number 100 31 15 10 12 10 8 5 6 3
Prime number 168 25 19 19 20 17 18 18 17 15
Mixing sequence 618 175 90 71 61 47 48 50 41 35
According to Euler, a natural number n is a numeri ideoni if it fulfills the following
condition. If a number p has a unique representation in the form px ny=+
22
with
relatively prime numbers x and y, then it is a prime number. Euler discovered the fact that
there are only 65 numeri ideoni, with n = 1848 the largest one. The proof of finiteness has
been given by Heilbronn and Chowla in 1934 (see Fellmann(1983), p. 37). Keith numbers or
repfigit numbers (replicating Fibonacci digits) have been introduced by Keith(1987) (see also
Pickover(1990/2000), Shiriff(1994), Keith(1994)). A Keith number is an n-digit integer N
with the following property. If a Fibonacci-like sequence (in which each term in the sequence
is the sum of the n previous terms) is formed, with the first n terms being the decimal digits
of the number N, then N itself occurs as a term in the sequence. For example, 47 is a Keith
number since it generates the sequence 4, 7, 11, 18, 29, 47. There are 71 Keith numbers up to
10
19
, as determined by Keith(1998). Moreover, it has been suggested that there is an infinite
number of Keith numbers. The story of Princeton numbers is found in Robbins(1991) (see
also Pickover(2000)). Lucky numbers and Ulam numbers are defined in Guy(1981), p. 60 (for
the latter ones see also Pickover(2000)). The Fibonacci numbers are well-known and need no
special reference. The n-th Bell number represents the number of ways to partition n things
into subsets (e.g. Aigner(1975), Jacobs(1983), Graham et al.(1989), Conway and Guy(1996)).
Prime numbers are ubiquitous and known to everybody. The mixing sequence represents the
aggregate of all previous integer sequences.
5
Table 3.2 : Fitting integer sequences to the Benford and generalized Benford distributions
Benford generalized Benford
Name of sequence MLE
$
c
ln L
2
p-value
ln L
2
p-value
Numeri ideoni 1.15087 127.76 2.59 95.7 127.71 2.52 92.5
Keith number 2.63445 142.49 9.21 32.4 141.98 7.72 35.8
Princeton number 2.81958 53.14 3.45 90.3 52.83 2.87 89.7
Lucky number 3.15802 83.96 7.69 46.4 82.78 5.16 64.0
Ulam number 3.48760 81.78 6.35 60.8 79.83 2.53 92.5
Fibonacci number 2.01802 199.22 1.03 99.8 199.22 1.02 99.4
Bell number 2.01106 200.39 3.07 93.0 200.39 3.07 87.9
Prime number 2.75660 389.05 45.02
410
5
387.04 37.03
46 10
4
.
Mixing sequence 2.52473 1277.65 15.55 4.93 1274.66 9.02 25.1
Taking into account the finiteness of the sequence of numeri ideoni, it is quite surpising how
well this sequence fits both first digit distributions. Among the Keith, Princeton, lucky and
Ulam numbers, the Princeton ones are the most likely to fit Benford’s law. Unfortunately, the
considered sequence is up to now only limited to a relatively small sample size. Comparing
p-values, the lucky and Ulam numbers are better fitted by the generalized Benford
distribution. The Fibonacci and Bell numbers are very close to Benford’s law. In fact, both
follow Benford’s law. For the Fibonacci sequence, this is shown in a number of papers
including Brown and Duncan(1970), Wlodarski(1971), Sentance(1973), Webb(1975),
Raimi(1976), Brady(1978), Kunoff(1987). An analytical proof that Bell numbers are Benford
distributed is given in Theorem 4.1. Both Benford distributions are rejected for the sequence
of primes. This statistical evidence proof is simpler than the analytical derivation proposed in
Diaconis(1977), p. 74. However, note that the sequence of primes is Benford distributed with
respect to other densities rather than with the usual natural density (e.g. Whitney(1972),
Schatte(1983), Cohen and Katz(1984)). Despite the observation that mixing sequences are
often Benford distributed (e.g. Benford(1938), Raimi(1969/85), Knuth(1969)), a fact proven
under some hypotheses by Hill(1995c), Theorem 3, it is perhaps surprising that our mixing
sequence is rejected at the 5% significance level. In contrast to this, the generalized Benford
law is accepted with a 25% p-value. Moreover, it is not clear how some sequences in Table
3.2 will behave by increasing the sample size. To clarify these and other facts, future research
is strongly required in this area.
4. The first digits of the Bell numbers satisfy Benford’s law.
To confirm the statistical evidence of Table 3.2, it is shown that the first digits of the Bell
numbers satisfy Benford’s law.
Theorem 4.1. The sequence of first digits of the Bell numbers are Benford distributed.
Proof. Recall that the Bell numbers satisfy the recurrence relationship b
n
k
b
nk
k
n
+
=
=
1
0
,
bb
12
12==, (e.g. Graham et al.(1989), p. 359). The sequence
{}
b
n
is Benford distributed if
6
and only if the sequence
{}
logb
n
is uniformly distributed modulo 1 (e.g. Diaconis(1977),
Theorem 1). To show the latter, consider first the sequence
x
ne
nn
n
nn
nn n
=
++
+⋅
−⋅
ln
ln ( )
ln
ln
1
2
1
1
12
,
which by Graham et al.(1989), p. 479 and 573, is an asymptotic estimate of
{}
lnb
n
. The
simpler sequence
y
ne
n
n
nn
nn n
=
+
+⋅
−⋅
ln
ln
ln
ln
1
2
1
1
satisfies the property
{}
lim
n
nn
xy
→∞
−=0 . In the following, we apply some useful results from
the theory of uniformly distributed sequences modulo 1 from Kuipers and Niederreiter(1974),
pp. 1-18. First, if two sequences
{}
x
n
and
{}
y
n
satisfy asymptotically a relation
{}
lim .
n
nn
x y const
→∞
−= , and if
{}
y
n
is uniformly distributed modulo 1, then
{}
x
n
is also
uniformly distributed modulo 1. Therefore, it suffices to show that
{}
y
n
/ ln10 is uniformly
distributed modulo 1. Second, one applies the criterion of Weyl(1916), which says that a
sequence
{}
a
n
is uniformly distributed modulo 1 if and only if
lim
log
N
ih a
n
N
N
e
n
→∞
=
=
1
0
2
1
π
for all hN
+
.
Third, to show this, we use Van der Corput’s estimate for trigonometric sums. Let a and b
be integers with
ab< , and let f be twice differentiable on
[
)
ab, with fx''( ) ≥>
0 or
fx''( ) ≤− <
0 for
[
)
xab , . Then
()
efbfa
if n
na
b
2
2
4
3
π
ρ
()
'( ) '( )
=
≤−++
.
Now, set fx hgx() ()/ln=⋅ 10 such that fn hy
n
() /ln=⋅ 10 with
()
()
gx x x x x x x x( ) ln ln ln ln ln=− + + +11
1
2
1
2
.
One has
() () ( )
[]
gxxxx xx'( ) ln ln ln=+++
21
1
221 ,
gx gx gx gx''( ) ( ) ( ) ( )=++
123
, with
7
()
gx
x
x
gx
x
x
gx
x
xx
122 3
2
2
21
2
2
1
2
2
1
() , ()
ln
,()
ln
ln
=
=⋅ =
+
+
.
We show that g
x
''( ) is decreasing for all
x
1 . This follows easily noting that
gx
x
x
13
1
0
'
()=
for
x
1, gx
x
x
22
1
0
'
()
ln
=
for
x
e ,
()
()
gx
xx
xx
3
2
1
2
71 2
1
0
'
()
ln (ln )
ln
=−
++
+
< , and gx gx
23
0
''
() ()+≤ for
x
1.
With abN==1, , it follows that for all
[]
xN 1, :
()
gx gN
N
N
N
N
NN
N
''( ) ''( )
ln ln
ln
≥≥++
+
+
1
2
2
1
2
2
1
1
2
2
2
,
which implies that fx hgx hN''( ) ''( ) /ln /=≥=>10 6 0
. Van der Corput’s estimate yields
()
()
e
h
NN
NN N
N
h
if n
n
N
2
1
2
10
1
2
1
21
24
6
3
π
()
ln
ln ln
ln
=
≤++
+
+
⋅+
,
which implies that
lim
()
n
if n
n
N
N
e
→∞
=
=
1
0
2
1
π
. Therefore, the asymptotically equivalent sequences
{}
y
n
/ln10 ,
{}
x
n
/ln10 and
{}
logb
n
are all uniformly distributed modulo 1. The result is
shown.
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Werner Hürlimann
Schönholzweg 24
CH-8409 Winterthur
Switzerland
e-mail : whurlimann@bluewin.ch
URL : www.geocities.com/hurlimann53
... Along this path one might mention papers from Leemis et al. [23], Engel and Leuenberger [8], and others, to recent contributions by Wójcik [44], [45]. Relying on [15], [18], our focus concentrates again on those parametric FDD's that extend and embed Benford's law. Three parametric extensions of Benford's archetype are considered. ...
... The author [15], Theorem 2.1, derives this one-parameter extension of Benford's law from a two-sided power random variable. For 1 ≥ c the corresponding FDD is given by ...
... . Although Stigler's law and the MEG data are not selected by the WLS criterion for any of the considered parametric extensions of Benford's law, there exist different probabilistic models of FDD's for which they will be selected. For this, the reader is refereed to the recent paper [15], which proposes a fine structure index for Benford's law. Finally, it is worthwhile to mention some other results related to the role GB might play for some real-world data sets. ...
Article
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As departures from Benford’s law have been observed in many scientific data sets, there is a theoretical need to understand such discrepancies. We argue that the use of parametric extensions to Benford’s law is appropriate and demonstrate this for several first significant digit distributions taken from theoretical scientific laws or extracted from real-world data sets.
... On the other hand, the first digit sequences of prime numbers follow a GB with size-dependent parameter 1 → α that converges necessarily to the uniform distribution, as motivated by Luque and Lacasa (2009). Alternative parametric generalizations of Benford's law have been considered by various authors including Hürlimann (2003/15a), Rodriguez (2004), Fu et al. (2007), equation (3), Grendar et al. (2007), etc. Concerning the present topic, Hürlimann (2009) finds numerical evidence that the sequence of primes might be Pareto Benford distributed, a first digit two-parameter probability distribution derived from the double Pareto in Reed (2001). In view of the promising and simple results by Luque and Lacasa (2009), it appears natural to refine their approach and extend it to prime powers. ...
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It is known that the first digits of prime numbers follow a generalized Benford law (GB) with size-dependent exponent that converges asymptotically to the uniform distribution. Based on two different statistics, we show the existence of size-dependent exponents that outperform in precision the optimal size-dependent exponent in Luque and Lacasa (2009) uniformly over some finite ranges. This result also holds for prime squares. Extending the approach to prime powers, different rates of convergence of the size-dependent GB’s to a GB with inverse power exponent are determined and compared. Furthermore, we introduce a criterion of counting compatibility, which indicates whether or not a given size-dependent GB that belongs to the first digits of some integer sequence is compatible with the asymptotic counting function of this sequence if it exists. We show the existence of a one-parametric size-dependent GB for the sequence of prime powers that is counting compatible with the prime number theorem and determine its optimal size-dependence. Finally, based on the theory of distribution functions of integer sequences, it is proved that the first digits of prime powers converge asymptotically to a GB with inverse power exponent. In particular, asymptotically as the power goes to infinity the sequences of prime powers obey Benford’s law.
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On the occasion of the 125-th anniversary of Newcomb's paper, a bibliography of academic work related to Benford's law from its year of origin 1881 to 2006 has been compiled.
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In THE BOOK OF NUMBERS, two famous mathematicians fascinated by beautiful and intriguing number patterns share their insights and discoveries with each other and with readers. John Conway is the showman, master of mathematical games and flamboyant presentations; Richard Guy is the encyclopedist, always on top of problems waiting to be solved. Together they show us why patterns and properties of numbers have captivated mathematicians and non-mathematicians alike for centuries. THE BOOK OF NUMBERS features Conway and Guy's favorite stories about all the kinds of numbers any of us is likely to encounter, and many others besides. "Our aim," the authors write, "is to bring to the inquisitive reader. . .an explanation of the many ways the word 'number' is used." They explore patterns that emerge in arithmetic, algebra, and geometry, describe these pattern' relevance both inside and outside mathematics, and introduce the strange worlds of complex, transcendental, and surreal numbers. This unique book brings together facts, pictures and stories about numbers in a way that no one but an extraordinarily talented pair of mathematician/writers could do.
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The accountant Nigrini remarked that in tables of data distributed according to Benford's law, the sum of all elements with first digit d (d = 1, 2,· ··, 9) is approximately constant. In this note, a mathematical formulation of Nigrini's observation is given and it is shown that Benford's law is the unique probability distribution such that the expected sum of all elements with first digits d1, · ··, dk is constant for every fixed k.
Chapter
Die vorliegende Darstellung will und kann keine Biographie, sondern sie soll — wie ihr Titel verrät — ein Essay sein. Dieser ist zweiteilig: der erste Teil bezieht sich mehr auf den äusseren Lebenslauf und den Charakter Eulers, der zweite auf sein wissenschaftliches Werk. Die Trennung beider Sphären ist nicht mehr als ein systematischer Behelf.