Distribution Of Digits Research Articles

The value of the last digit: statistical fraud detection with digit analysis

Digit distributions are a popular tool for the detection of tax payers’ noncompliance and other fraud. In the early stage of digital analysis, Nigrini and Mittermaier (A J Pract Theory 16(2):52–67, 1997) made use of Benford’s Law (Benford in Am Philos Soc 78:551–572, 1938) as a natural reference distribution. A justification of that hypothesis is only known for multiplicative sequences (Schatte in J Inf Process Cyber EIK 24:443–455, 1988). In applications, most of the number generating processes are of an additive nature and no single choice of ‘an universal first-digit law’ seems to be plausible (Scott and Fasli in Benford’s law: an empirical investigation and a novel explanation. CSM Technical Report 349, Department of Computer Science, University of Essex, http://cswww.essex.ac.uk/technical-reports/2001/CSM-349.pdf , 2001). In that situation, some practioneers (e.g. financial authorities) take recourse to a last digit analysis based on the hypothesis of a Laplace distribution. We prove that last digits are approximately uniform for distributions with an absolutely continuous distribution function. From a practical perspective, that result, of course, is only moderately interesting. For that reason, we derive a result for ‘certain’ sums of lattice-variables as well. That justification is provided in terms of stationary distributions.

Stigler’s approach to recovering the distribution of first significant digits in natural data sets

Benford’s Law can be seen as one of the many first significant digit (FSD) distributions in a family of monotonically decreasing distributions. We examine the interrelationship between Benford and other monotonically decreasing distributions such as those arising from Stigler, Zipf, and the power laws. We examine the theoretical basis of the Stigler distribution and extend his reasoning by incorporating FSD first-moment information into information-theoretic methods. We present information-theoretic methods as a way to describe, connect, and unify these related distributions and thereby extend the reach of Benford’s Law and FSD research more generally.

Order Statistics and Benford′s Law

Fix a base B > 1 and let ζ have the standard exponential distribution; the distribution of digits of ζ base B is known to be very close to Benford′s law. If there exists a C such that the distribution of digits of C times the elements of some set is the same as that of ζ, we say that set exhibits shifted exponential behavior base B. Let X1, …, XN be i.i.d.r.v. If the Xi′s are Unif, then as N → ∞ the distribution of the digits of the differences between adjacent order statistics converges to shifted exponential behavior. If instead Xi′s come from a compactly supported distribution with uniformly bounded first and second derivatives and a second‐order Taylor series expansion at each point, then the distribution of digits of any Nδ consecutive differences and all N − 1 normalized differences of the order statistics exhibit shifted exponential behavior. We derive conditions on the probability density which determine whether or not the distribution of the digits of all the unnormalized differences converges to Benford′s law, shifted exponential behavior, or oscillates between the two, and show that the Pareto distribution leads to oscillating behavior.

Risk factor measurement quality in primary care routine data was variable but nondifferential between individuals

Objective To retrospectively assess the quality of cardiovascular disease risk factor measurements in routinely collected data and to examine whether there are systematic differences in measurement quality between individuals of different socioeconomic status, and changes in measurement quality over time. Study Design and Setting Comparison of last digit preference in risk factor measurement between a “routine” data set (related to a primary care–based UK cardiovascular risk factor screening programme) and relevant prospective epidemiological surveys (Health Survey for England [HSE] and WHO MONICA). For the routine data set, the records of 37,161 women and 33,977 men aged 35–60 years with a first screening episode during the 11-year period 1989–1999 were analyzed. Results Last digits of height, weight, and total cholesterol measurements in the routine data set did not exhibit a digit preference pattern, although the degree of random measurement error was greater compared with epidemiological survey data. The last digits of systolic and diastolic blood pressure (BP) measurements exhibited a strong last digit preference pattern for “0”—comprising 63.1% and 67.3% of all observations in men and women, respectively. Except for diastolic BP in men, last digit distribution patterns were not associated with participant's socioeconomic status and showed no change over time. Conclusion It may be feasible to study changes over time in cardiovascular disease risk factor levels in different socioeconomic groups using routine data sets; however, prior critical examination of measurement quality is necessary.

Analysing digits for portfolio formation and index tracking

A general methodology is proposed for using digit distributions as an approach to examining arbitrary datasets. With the Newcomb–Benford law as a starting point, a more general framework for digital analysis is developed. A new measure is proposed based on this framework, namely the Digital-Fit Factor (DFF). The use of index comparison on the S&P500 and the Dow Jones Industrial Average is demonstrated. The DFF is then used to construct portfolios and measure their performance compared with that of the index. The average returns using the measure exceed the index composition by 6–14 percentage points per year by being more stable at the same time. Furthermore, these measures require only a very small proportion of the available information and are thus very efficient.

Assessing the Accuracy of Self-Reported Data: an Evaluation of the Toxics Release Inventory

Self-reported regulatory data are hard to verify. This article compares air emissions reported by plants in the Toxics Release Inventory with chemical concentration levels measured by EPA pollution monitors. We find that the large drops in air emissions reported by firms in the TRI are not always matched by similar reductions in measured concentrations from EPA monitors. When the first digits of the monitored chemical concentrations follow a monotonically decreasing distribution, we expect (via Benford's Law) a similar distribution of first digits for the TRI data. For lead and nitric acid the self-reported data do not follow the expected first digit pattern. This suggests that for these two heavily regulated chemicals plants are not reporting accurate estimates of their air emissions.

Examining data accuracy and authenticity with leading digit frequency analysis

This paper proposes and demonstrates that leading digit frequency analysis offers a potential framework to examine the accuracy and authenticity of data values in numerical data sets. Leading digit frequency analysis maintains that the frequencies of leading digits in an original and unmanipulated data set follow a predictable distribution as suggested by Benford’s law. The results of an experiment with random numbers revealed distinct differences between the observed leading digit distribution in the random numbers and the expected distribution as suggested by Benford’s law. This suggests that leading digit frequency analysis offers a useful procedure to evaluate data accuracy and authenticity and offers several useful managerial implications.

On the nature of Benford's Law

We study multiplicative and affine sequences of real numbers defined by N(j+1)=ζ(j)N(j)+η(j) , where { ζ( j)} and { η( j)} are sequences of positive real numbers (in the multiplicative case η( j)=0 for all j). We investigate the conditions under which the leading digits k of { N( j)} have the following probability distribution, known as Benford's Law, P( k)=log 10(( k+1)/ k). We present two main results. First, we show that contrary to the usual assumption in the literature, { ζ( j)} does not necessarily need to come from a chaotic or independent random process for Benford's Law to hold. The multiplicative driving force may be a deterministic quasiperiodic or even periodic forcing. Second, we give conditions under which the distribution of the first digits of an affine process displays Benford's Law. Our proofs use techniques from ergodic theory.

Digit distribution of proper digital nerve action potential.

Antidromic sensory nerve action potential testing is well characterized and commonly used to assess the sensory component of the upper limb median and ulnar nerves. The final terminal segments of these nerves are the proper digital nerves. Ring recording electrodes are commonly used to detect the proper digital nerves' antidromic responses. Attempts to record the separate contributions of individual digital nerves along the lateral aspects of each finger, using small surface electrodes, is shown to be unreliable for determining the integrity of a single terminal digital branch. We found between 50% to 77% of the stimulated terminal branch's response amplitude when recorded at electrodes positioned over the nonstimulated branch located 180 degrees from the activated terminal branch. Detecting a single terminal nerve response was achieved by using the fourth digit and the second digit with one of the second digit's branches neurophysiologically blocked by local anesthetic. The volume-conducted response from the opposite side of the finger resulted in this relatively large recorded response, which remains within the range of reference values precluding the simple use of antidromic techniques to assess injury to a single proper digital nerve. Techniques are proposed to avoid such pitfalls and to assess most accurately the desired response.

Explaining the uneven distribution of numbers in nature: the laws of Benford and Zipf

The distribution of first digits in numbers series obtained from very different origins shows a marked asymmetry in favor of small digits that goes under the name of Benford's law. We analyze in detail this property for different data sets and give a general explanation for the origin of the Benford's law in terms of multiplicative processes. We show that this law can be also generalized to series of numbers generated from more complex systems like the catalogs of seismic activity. Finally, we derive a relation between the generalized Benford's law and the popular Zipf's law which characterize the rank order statistics and has been extensively applied to many problems ranging from city population to linguistics.

Digit preference in year at menopause: Data from the study of women's health across the nation

PURPOSE: To assess possible digit preference in self-reported year at natural menopause, and to determine whether equal proportions is an appropriate reference distribution.METHODS: Data are from the cross-sectional telephone interview from the Study of Women's Health Across the Nation (SWAN), a multi-site, multi-ethnic study of women aged 40–55. Analyses included 2151 naturally menopausal respondents, who were asked the year their periods stopped. Using a chi-square test, we compared the distribution of the terminal digit for year of final menstrual period (FMP) to an equal proportions (EP) distribution assigning 10% probability of each of the 10 digits. Departures from EP, however, may be accurate and may reflect the observed age distribution of subjects rather than digit preference. Thus, we stratified by year of interview to determine if the distribution was the same across years—suggesting digit preference—or if it varied systematically. We then estimated an expected distribution of terminal digit for year of FMP, using prospectively collected data (not subject to digit preference or recall bias) from the Massachusetts Women's Health Study, applied to the SWAN age distribution. The SWAN terminal digit distribution was compared with this expected distribution.RESULTS: Terminal digit for year at FMP exhibited a strong departure from EP (χ2 = 841.60, p < 0.001), with highest frequencies for digits 4 and 5. Stratifying by year of interview, the distribution was shifted one digit in 1997 compared with 1996, consistent with unbiased reporting. Using the expected distribution as the reference reduced the chi-square statistic by a factor of 7 (χ2 = 119.51, p < 0.001).CONCLUSIONS: The distribution of terminal digits for reported year at FMP was far from uniform, but departures from EP were consistent with results expected from an independent prospective study. These results suggest that EP may not be an appropriate reference for studies of digit preference, particularly those with a restricted age range.

Singular and fractal properties of distributions of random variables digits of polybasic representations of which a form homogeneous Markov chain

We study the fractal properties of distributions of random variables digits of polybasic Q-representations (a generalization of n-adic digits) of which form a homogeneous Markov chain in the case where the matrix of transition probabilities contains at least one zero.

The Hausdorff dimension of a class of fractals, which are determined by the distribution of dyadic digits

An explicit formula is given for the Hausdorff dimension (HD) of certain Borel sets.

Using Benford’s law and neural networks as a review procedure

Introduces a new analytical review procedure that measures the degree to which a data set’s digit distribution deviates from a Benford digit distribution. This deviation can indicate potential manipulation and can be used to signal the need for further audit testing. An artificial neural network is used to distinguish between “normal” and “manipulated” financial data. The results show that if data have been contaminated (at a 10 per cent level or more) a Benford analytical review procedure will detect this 68 per cent of the time. If the data are not contaminated, the test will indicate that the data are “clean” 67 per cent of the time. Because analytical review procedures are not used in isolation, these results probably understate the effectiveness and potential of a digits‐based analytical review procedure. This procedure’s fraud detection results compare favorably to traditional analytical review procedures. Importantly, its unique analysis procedure allows it to complement traditional analytical review procedures. A key limitation of this study is that it uses simulated data, rather than actual data. Such an enhancement will be a critical step in future research. This method appears to have potential merit and provides many opportunities for new research.

The relationship between Zipf's law and the distribution of first digits

Summary Zipf 's experimental law states that, for a given large piece of text, the product of the relative frequency of a word and its order in descending frequency order is a constant, shown to be equal to 1 divided by the natural logarithm of the number of different words. It is shown to be approximately equal to Benford's logarithmic distribution of first significant digits in tables of numbers. Eleven samples allow comparison of observed and theoretical frequencies.

Distribution of consecutive digits in theq-ary expansions of some subsequences of integers

Asymptotic normality is established for consecutive digits in the q-ary expansions of some subsequences of integers.

Uniform occurrence of digits for folded and mixture distributions on finite intervals

Historically, the occurrences of digits in a set of observed values of a random variable are of interest in theoretical and applied statistics for various reasons including rounding errors and statistical forensics. A probabilistic approach to such occurrences is needed in order to draw conclusions about the phenomena represented by these observed values. In this paper, we study certain structural properties of finite decimal expansions of real numbers from finite intervals within the framework of 10 k -folded distributions defined on such intervals. To provide a probabilistic structure for the occurrence of digits we obtain characterization results for the rectangular and folded-rectangular distributions. A brief discussion of the use of 10 k -bin histogram distributions to determine the approximate distribution of digits, along with two examples, illustrate our result.

Data fabrication: Can people generate random digits?

Many people have difficulty in generating random numbers. This difficulty suggests that potentially fabricated numbers encountered in investigations of scientific misconduct be examined for nonrandom behavior. The present paper shows that even with a conscious effort to construct random digits, many subjects are unable to produce digits with a uniform distribution. For this study, subjects were directed to try to produce random digits in three places in order to fabricate a series of “pick 3”; lottery numbers. Subjects were most successful at producing a random (uniform) distribution of digits for the leftmost place; however, success at one place was not associated with success at another. In addition, subjects did not select all digits with equal frequency. Of 8,280 digits chosen in this study, the order from most to least chosen was 1, 2, 3, 6, 4, 9, 7, 0, 8, 5. Finally, no strong correlations among subjects’ digit choices were found. The conscious effort by these subjects to produce random digits stands in contrast with the usual case of data fabrication in which the fabricator must devote a conscious effort to choose leftmost digits so the number has the magnitude desired and pays little or no attention to the fact that the rightmost digits should be random. The results of the present paper indicate that even if a data‐fabricator were aware that error digits would be examined for uniformity, success in constructing uniform error distributions is not guaranteed. The difficulty that people have in creating random error digits supports the utility of examining such digits in investigations of scientific misconduct.

Benford’s law and physical constants: The distribution of initial digits

John Burke and Eric Kincanon Citation: American Journal of Physics 59, 952 (1991); doi: 10.1119/1.16838 View online: http://dx.doi.org/10.1119/1.16838 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/59/10?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in Do dynamical systems follow Benford’s law? Chaos 10, 331 (2000); 10.1063/1.166498 The first digit distribution law Am. J. Phys. 55, 584 (1987); 10.1119/1.15088 A constant force generator for the demonstration of Newton’s second law Am. J. Phys. 51, 344 (1983); 10.1119/1.13245 A critical review of Henry’s law constants for chemicals of environmental interest J. Phys. Chem. Ref. Data 10, 1175 (1981); 10.1063/1.555654 Laser‐initiated chemical reactions: Cl+H2S→HCl+HS: Rate constant, product energy distribution, and directdetection of a chain mechanism J. Chem. Phys. 69, 839 (1978); 10.1063/1.436599

On the numbers of things and the distribution of first digits

I answer the question ‘‘Why are there more small things in the world than large things?’’ in terms of a probabilisitc model of partitioning a conserved quantity. Benford’s empirical rule for lists of numbers, that the proportion of numbers with first digit m is log10(m+1)−log10 m, is an exact consequence of the model.