Generalizing Benford's Law Using Power Laws: Application to Integer Sequences
Many distributions for first digits of integer sequences are not Benford. A simple method to derive parametric analytical extensions of Benford's law for first digits of numerical data is proposed. Two generalized Benford distributions are considered, namely, the two-sided power Benford (TSPB) distribution, which has been introduced in Hürlimann (2003), and the new Pareto Benford (PB) distribution. Based on the minimum chi-square estimators, the fitting capabilities of these generalized Benford distributions are illustrated and compared at some interesting and important integer sequences. In particular, it is significant that much of the analyzed integer sequences follow with a high -value the generalized Benford distributions. While the sequences of prime numbers less than 1000, respectively, 10 000 are not at all Benford or TSPB distributed, they are approximately PB distributed with high -values of 93.3% and 99.9% and reveal after a further deeper analysis of longer sequences a new interesting property. On the other side, Benford's law of a mixing of data sets is rejected at the 5% significance level while the PB law is accepted with a 93.6% -value, which improves the -value of 25.2%, which has been obtained previously for the TSPB law.
Since Newcomb  and Benford  it is known that many numerical data sets follow Benford’s law or are closely approximated by it. To be specific, if the random variable , which describes the first significant digit in a numerical table, is Benford distributed, then
Mathematical explanations of this law have been proposed by Pinkham , Cohen , Hill [5–9], Allart , Janvresse and de la Rue , and Kossovsky . The latter author has raised some conjectures, which have been proved in some special cases by Jang et al. . Other explanations of the prevalence of Benford’s law exist. For example, Miller and Nigrini  obtain it through the study of products of random variables and Kafri  through the maximum entropy principle. In the recent years an upsurge of applications of Benford’s law has appeared, as can be seen from the compiled bibliography by Hürlimann  and the recent online bibliography by Berg and Hill . Among them one might mention Judge and Schechter , Judge et al. , and Nigrini and Miller . As in the present paper, the latter authors also consider power laws.
Hill  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.  and Engel and Leuenberger . Some survival distributions, which satisfy exactly Benford’s law, are known. However, there are not many simple analytical distributions, which include as special case Benford’s law. Combining facts from Leemis et al.  and Dorp and Kotz  such a simple one-parameter family of distributions has been considered in Hürlimann . In a sequel to this, a further generalization of Benford’s law is considered.
It is important to note that many distributions for first digits of integer sequences are not Benford but are power laws or something close. Thus there is a need for statistical tests for analyzing such hypotheses. In this respect the interest of enlarged Benford laws is twofold. First, parametric extensions may provide a better fit of the data than Benford’s law itself. Second, they yield 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 through our application to integer sequences.
2. Generalizing Benford’s Distribution
If denotes a random lifetime with survival distribution , then the value of the first significant digit in the lifetime has the probability distribution
Alternatively, if denotes the integer-valued random variable satisfying
then the first significant digit can be written in terms of and as
where denotes the greatest integer less than or equal to . In particular, if the random variable is uniformly distributed as , then the first significant digit is exactly Benford distributed. Starting from the uniform random variable or the triangular random variable with probability density function if and if , one shows that the random lifetime generates the first digit Benford distribution (Leemis et al. [21, Examples 1 and 2]).
A simple parametric distribution, which includes as special cases both the above uniform and triangular distributions, is the twosided power random variable considered in Dorp and Kotz  with probability density function
If then , and if then . This observation shows that the random lifetime will generate first digit distributions closely related to Benford’s distribution, at least if is close to 1 or 2.
Theorem 2.1. Let be the twosided power random variable with probability density function and let the integer-valued random variable satisfy . Then the first digit random variable has the one-parameter twosided power Benford (TSPB) probability density function
Proof. This has been shown in Hürlimann .
3. From the Geometric Brownian Motion to the Pareto Benford Law
Another interesting distribution, which also takes the form of a twosided power law, is the double Pareto random variable considered in Reed  with probability density function
Recall the stochastic mechanism and the natural motivation, which generates this distribution. It is often assumed that the time evolution of a stochastic phenomena involves a variable but size independent proportional growth rate and can thus be modeled by a geometric Brownian motion (GBM) described by the stochastic differential equation
where is the increment of a Wiener process. Since the proportional increment of a GBM in time has a systematic component and a random white noise component , GBM can be viewed as a stochastic version of a simple exponential growth model. The GBM has long been used to model the evolution of stock prices (Black-Scholes option pricing model), firm sizes, city sizes, and individual incomes. It is well known that empirical studies on such phenomena often exhibit power-law behavior. However, the state of a GBM after a fixed time follows a lognormal distribution, which does not exhibit power-law behavior.
Why does one observe power-law behavior for phenomena apparently evolving like a GBM? A simple mechanism, which generates the power-law behavior in the tails, consists to assume that the time of observation itself is a random variable, whose distribution is an exponential distribution. The distribution of with fixed initial state is described by the double Pareto distributio n with density function (3.1), where , and are the positive roots of the characteristic equation
where is the parameter of the exponentially distributed random variable . Setting one obtains the following generalized Benford distribution.
Theorem 3.1. Let be the double Pareto random variable with probability density function
Let the integer-valued random variable satisfy . Then the first digit random variable has the two-parameter Pareto Benford (PB) probability density function
Proof. The probability density function of is given by It follows that the first significant digit of , namely, , has probability density Making the change of variable , one obtains (3.5) as follows:
One notes that setting and letting goes to infinity, the Pareto Benford distribution converges to Benford’s law. Other important paper, which links Benford’s law to GBMs’ law on the one side, is Kontorovich and Miller  and to Black-Scholes’ law on the other side is Schürger . Another law, which includes as a special case the Benford law, is the Planck distribution of photons at a given frequency, as shown recently by Kafri [28, 29].
4. Fitting the First Digit Distributions of Integer Sequences
Minimum chi-square estimation of the generalized Benford distributions is straightforward by calculation with modern computer algebra systems. The fitting capabilities of the new distributions are illustrated at some interesting and important integer sequences. The first digit occurrences of the analyzed integer sequences are listed in Table 1. The minimum chi-square estimators of the generalized distributions as well as an assumed summation index for the infinite series (3.5) are displayed in Table 2. Statistical results are summarized in Table 3. For comparison we list the chi-square values and their corresponding -values. The obtained results are discussed.
The definition, origin, and comments on the mathematical interest of a great part of these integer sequences have been discussed in Hürlimann . Further details on all sequences can be retrieved from the considerable related literature. The mixing sequence represents the aggregate of the integer sequences considered in Hürlimann .
All of the 19 considered integer sequences are quite well fitted by the new PB distribution. For 14 sequences the minimum chi-square is the smallest among the three comparative values and in the other 5 cases its value does not differ much from the chi-square of the TSPB distribution ( bold cells in Table 3 and Table 5).
A strong numerical evidence for the Benford property for the Fibonacci, Bell, Catalan, and partition numbers is observed (corresponding italic cells in Tables 2 and 3). In particular, the values of the parameters of the BP distribution for the Fibonacci sequence are close to 1 and , which means that the BP distribution is almost Benford as remarked after Theorem 3.1. It is well known that the Fibonacci sequence is Benford distributed (e.g., Brown and Duncan , Wlodarski , Sentance , Webb , Raimi (1976),  Brady  and Kunoff ).The same result for Bell numbers has been derived formally in Hürlimann [24, Theorem 4.1]. More generally, a proof that a generic solution of a generic difference equation is Benford is found in Miller and Takloo-Bighash  (see also Jolissaint [38, 39]). Results for squares and cubes are also obtained. Recall that the exact probability distribution of the first digit of th integer powers with at most digits is known and asymptotically related to Benford’s law (e.g., Hürlimann ). The fit of the PB distribution is very good when restricted to finite sequences but breaks down for longer sequences. A further remarkable result is that Benford’s law of the mixing sequence is rejected at the 5% significance level while the PB law is accepted with a 93.6% -value, which improves the -value of 25.2% obtained for the TSPB law in Hürlimann .
The sequence of primes merits a deeper analysis. The Benford property for it has long been studied. Diaconis (1977)  shows that primes are not Benford distributed. However, it is known that the sequence of primes is Benford distributed with respect to other densities rather than with the usual natural density [42–44]. According to Serre [45, Page76], , Bombieri has noted that the analytical density of primes with first digit 1 is , and this result can be easily generalized to Benford behavior for any first digit. Table 3 shows that the primes less than 1,000 respectively 10 000 are not at all Benford or TSPB distributed, but they are approximately PB distributed with high -values of 93.3% and 99.9%. Does this statistical result reveal a new property of the prime number sequence? To answer this question it is necessary to take into account longer sequences and look at other cutoffs than for an integer k . Our calculations show that among those prime sequences below for fixed k there is exactly one sequence with minimum chi-square value with an optimal cutoff at a prime with first digit 9. Tables 4 and 5 summarize our results for the primes up to . Besides the PB best fit with minimum chi-square we also list the PB “linear best” fit obtained from the PB best fit by taking a linear decreasing number of primes between those with the same number of primes with first digit 1 and 9 as in the PB best fit. Though the P-value goes to zero very rapidly the ratio of the minimum chi-square value to the sample size is more stable. For the PB linear best fit this goodness-of-fit statistic, which is also considered in Leemis et al. , even decreases and indicates therefore that the first digits of the prime number sequence might be distributed this way. For this it remains to test using more powerful computing whether the mentioned property still holds for even longer sequences of primes. One observes that the best fit parameters as the sample size increases to infinity are quite stable and increase only slightly.
Finally, it might be worthwhile to mention another recent intriguing result by Kafri , which shows that digits distribution of prime numbers obeys the Planck distribution, which is another generalized Benford law as already mentioned at the end of Section 3.
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