Benfords Law Research Articles

The Anomalous Behavior of Stock Prices on the Canadian Securities Exchange

This paper examines the conformity of the distribution of stock prices on the Canadian Securities Exchange (CSE) to that theorized by Benford’s Law (BL). BL follows a logarithm law such that the leading digits have a higher probability of lower numbers such as 1, 2, or 3 versus higher numbers like 7, 8, or 9. This analysis can be used to detect fraudulent manipulation of stock prices. Previous research has applied BL to a broad range of data such as cost data, atomic weights, river areas, and populations, as well as stock prices. After collecting stock prices on the CSE, the number count for each digit was compared to the expected number given the frequencies posited by BL. A chi-square test was employed to determine statistical significance. The first digit was found to adhere to BL; however, the second digit was not congruent with that predicted by BL. There is an indication of possible manipulation on this stock exchange. These mixed results are consistent with the empirical evidence of other researchers. This evidence is relevant to auditors, shareholders, financial analysts, investment managers, government, and the CSE.

Statistical Analysis of Covid-19 Outbreak with Benford’s Law

The coronavirus disease first identified in mid-December 2019 in Wuhan, China is an ongoing pandemic and the virus has spread around the world. As of 13 March 2020, the number of new cases started to increase significantly in Europe, and Europe was considered as the new center of the Covid-19 pandemic as announced by the WHO. Confirmed case rate (CCR), computed from the numbers of confirmed cases over numbers of tests of the countries can be used to confirm the quality of the numbers, and to detect the manipulation for health surveillance systems of the countries for managing the situation by testing whether or not follow Benford’s Law (BL). The main aim of this study is to test CCRs of the countries in Europe by BL to detect the data qualities and to monitor the manipulations, which can help to take precautions for the health surveillance systems of the countries.

Study on the effect of sample size on type I error, in the first, second and first-two digits excessmad tests

The first-two digits ExcessMAD test was created in 2016, allowing to evaluate whether a certain data set conforms to Benford’s Law (BL). The purpose of this study is to explore some questions that remained open: develop the exact and approximate mathematical formulation of the first and second digit ExcessMAD test and study the type I error of these tests when applied to different sample sizes conforming to BL and to the uniform distribution, due to its wide use in accounting data. The importance of this study is to make available to accountants, auditors and researchers the first and second digit ExcessMAD tests, which will make it possible to conduct further investigations involving BL, especially for smaller samples. In addition, the relevance of the type I error analysis stems from the reduction of unnecessary additional studies for the investigation of non-conformity, in the case of the erroneous rejection of the null hypothesis stated as conforming to BL. The application of the second digit ExcessMAD test in the uniform distribution reveals that the close proximity between the uniform and BL distributions can lead to misinterpretations. Based on the exact and approximate mathematical formulations of the three ExcessMAD tests and the use of the Monte Carlo simulation technique, samples were generated in accordance with the BL and uniform distributions, with sizes between 100 and 3,500 elements, which allowed the study of type I error and the comparison of the tests applied to those distributions. This paper seeks to cover three gaps in the literature on ExcessMAD tests. In the previous studies, the following approaches were not found: the exact and approximate mathematical formulation of the first and second digit ExcessMAD tests; the analysis of type I error in these tests and the comparison of such results in the BL and Uniform distributions.

Benford’s Law for Telemetry Data of Wildlife

Benford’s law (BL) specifies the expected digit distributions of data in social sciences, such as demographic or financial data. We focused on the first-digit distribution and hypothesized that it would apply to data on locations of animals freely moving in a natural habitat. We believe that animal movement in natural habitats may differ with respect to BL from movement in more restricted areas (e.g., game preserve). To verify the BL-hypothesis for natural habitats, during 2015–2018, we collected telemetry data of twenty individuals of wild red deer from an alpine region of Austria. For each animal, we recorded the distances between successive position records. Collecting these data for each animal in weekly logbooks resulted in 1132 samples of size 65 on average. The weekly logbook data displayed a BL-like distribution of the leading digits. However, the data did not follow BL perfectly; for 9% (99) of the 1132 weekly logbooks, the chi-square test refuted the BL-hypothesis. A Monte Carlo simulation confirmed that this deviation from BL could not be explained by spurious tests, where a deviation from BL occurred by chance.

Composite test inclusive of Benford’s law, noise reduction and 0–1 test for effective detection of chaos in rotor–stator rub

Segregating noise from chaos in dynamic systems has been one of the challenging work for the researchers across the globe due to their seemingly similar statistical properties. Even the most used tools such 0-1 test and Lyapunov exponents fail to distinguish chaos when signal is mixed with noise. This paper addresses the issue of segregating the dynamics in a rotor-stator rub system when the vibrations are subjected to different levels of noise. First, the limitation of 0-1 test in segregating chaos from signal mixed with noise has been established. Second, the underexplored Benfords Law and its application to the vibratory dynamical rotor-stator rub system has been introduced for the first time. Using the Benfords Law Compliance Test (BLCT), successful segregation of not only noise from chaos but also very low Signal to Noise Ratio (SNR) signals which are mainly stochastic has been achieved. The Euclidean Distance concept has been used to explore the scale-invariant probability distribution of systems that comply with Benfords Law to separate chaos from noise. Moreover, for moderate bands of noise in signals, we have shown that the Schreibers Nonlinear Noise Reduction technique works effectively in reducing the noise without damaging the dynamic properties of the system. Combining these individual layers (0-1 Test, BLCT and Noise reduction) on a rotor system, a Decision Tree based method to effectively segregate noise from chaos and identify the correct dynamics of any system with time series data set has been proposed.

Authoritarian regimes' propensity to manipulate Covid-19 data: a statistical analysis using Benford's Law

ABSTRACT There have been numerous claims that some countries do not reveal their reported cases of COVID-19 truthfully. Some studies suggest that autocratic countries tend to manipulate their reported figures. To test these claims, the article uses Benford’s law (BL) to detect anomalies in the reported numbers of COVID-19. Many studies have used BL to test the reliability of the data recorded by countries and conclude that the data follows BL in most cases. This study applies BL to a set of 34 countries using Pearson’s χ2, Kuiper and mean absolute deviation tests to analyse the daily reported cases; then, correlating the tests results with four freedom indices using ordinary least square. The aim of the article is to demonstrate how countries with low freedom scores are likely to manipulate their COVID-19 reporting, whereas countries with high freedom scores do not show a manipulation in their data. The study finds that all indices show strong correlation to the degree of data manipulation, and with the World Press Freedom index showing the strongest relationship.

IS BENFORD’S LAW EFFECTIVE IN FRAUD DETECTION FOR EXPENSE CYCLE?

Despite measures taken by firms to prevent fraud, it has been found in recent studies that losses derived from fraudulent activities are increasing on a global basis. International standards on auditing do not define which analytical approaches and technological tools to be used in performing audit. Decisions are left on the auditor’s judgment. Auditors try to use digital techniques to deal with mass data sets generated by firms. Academic research may mislead practitioners as controversial outcomes exist in literature concerning empirical research. Benford’s Law (BL) is one of the methods used frequently in digital analysis. Although some researchers defend the use of BL in audit, especially in fraud detection, this paper disputes its effectiveness for expense cycle. Different firm’s data are tested to conclude that the use of BL is not appropriate for expense items. The reasons of this deficiency are explained in this paper.

COVID-19, flattening the curve, and Benford’s law

For many countries attempting to control the fast-rising number of coronavirus cases and deaths, the race is on to “flatten the curve,” since the spread of coronavirus disease 2019 (COVID-19) has taken on pandemic proportions. In the absence of significant control interventions, the curve could be steep, with the number of COVID-19 cases growing exponentially. In fact, this level of proliferation may already be happening, since the number of patients infected in Italy closely follows an exponential trend. Thus, we propose a test. When the numbers are taken from an exponential distribution, it has been demonstrated that they automatically follow Benford’s Law (BL). As a result, if the current control interventions are successful and we flatten the curve (i.e., we slow the rate below an exponential growth rate), then the number of infections or deaths will not obey BL. For this reason, BL may be useful for assessing the effects of the current control interventions and may be able to answer the question, “How flat is flat enough?” In this study, we used an epidemic growth model in the presence of interventions to describe the potential for a flattened curve, and then investigated whether the epidemic growth model followed BL for ten selected countries with a relatively high mortality rate. Among these countries, South Korea showed a particularly high degree of control intervention. Although all of the countries have aggressively fought the epidemic, our analysis shows that all countries except for Japan satisfied BL, indicating the growth rates of COVID-19 were close to an exponential trend. Based on the simulation table in this study, BL test shows that the data from Japan is incorrect.

Usability of the Benford’s law for the results of least square estimation

Benford’s law (BL), also known as the first-digit or significant-digit law, is an intriguing pattern in data sets, considers the frequency of occurrence of the first digits, which are not uniformly distributed as might be expected, conversely follow a specified theoretical distribution. According to BL, the occurrence of first non-zero digit in a numerical data, which is generated or found in nature, depends on a logarithmic distribution. Least square estimation (LSE) method is mostly preferred for the estimation of the unknown parameters from different types of geodetic data. The residuals and the normalized residuals of the LSE method, which follow normal distribution and expected values of them are zero are used in outlier detection problem. In this study, BL is investigated for residuals and the normalized residuals estimated from LSE method. Three types of geodetic data are used: (1) simulated regression models, (2) global positioning system (GPS) data, (3) leveling network. The first group data sets are simulated based on linear regression and univariate models and each simulated group is generated for a number of 100, 1000, and 10,000 samples. To generate second group, an international global navigation satellite system (GNSS) service (IGS) station data (ISTA) is processed by kinematic PPP approach using GIPSY OASIS II v6.4 software. Here, the observation duration of GPS data is 4 days. For the last data, a leveling network with 55 points involving 110 observations of height differences is simulated. BL has been applied to the residuals (v) and normalized residuals (w) estimated from LSE method. Goodness-of-fit test has been implemented to determine whether a population has a specified BL distribution or not. This test is based on how good a fit we have between the frequency of occurrence of residuals and normalized residuals in an observed sample and the expected frequencies obtained from the hypothesized distribution. The results depending on the statistical test show that each data set (residuals and normalized residuals) used in this study follows BL.

Benford’s law and first letter of words

A universal First-Letter Law (FLL) is derived and described. It predicts the percentages of first letters for words in novels. The FLL is akin to Benford’s law (BL) of first digits, which predicts the percentages of first digits in a data collection of numbers. Both are universal in the sense that FLL only depends on the numbers of letters in the alphabet, whereas BL only depends on the number of digits in the base of the number system. The existence of these types of universal laws appears counter-intuitive. Nonetheless both describe data very well. Relations to some earlier works are given. FLL predicts that an English author on the average starts about 16 out of 100 words with the English letter ‘t’. This is corroborated by data, yet an author can freely write anything. Fuller implications and the applicability of FLL remain for the future.

Using Benford’s law on the seismic reflectivity analysis

Benford’s law (BL) is a mathematical theory of leading digits. This law predicts that the distribution of first digits of real-world observations is not uniform and follows a trend in which measurements with a lower first digit (1, 2, …) occur more frequently than those with higher first digits (…, 8, 9). A data set from earth’s geomagnetic field, the estimated time in years between reversals of earth’s geomagnetic field, the seismic P-wave speed of earth’s mantle below the southwest Pacific, and other geophysical data obey the BL. Although there are other statistical methods for analyzing a data set, we test, for the first time, the analysis of the seismic reflectivity through the Benford distribution point of view. We applied the BL on real reflectivity data from two wells from the Penobscot field and another two from the Viking Graben field. In both data sets, the reflectivity was in conformity with the BL. Moreover, after analyzing the effect of sonic and density logs despiking on Benford’s distribution through the BL, we found an optimum coefficient for the despiking process, which was a common procedure used to edit the well-log data before its use on reservoir studies.

Analysis of the dynamic characteristics of air-water two-phase flow in small channel based on multi-scale normalized Benford probability distribution

In gas and liquid two-phase flow, different flow patterns show their unique nonlinear dynamic characteristics due to the complex interaction between them. To further reveal the underlying dynamic characteristics of this physical phenomenon, this paper presents a new model for the analysis of small channel gas-liquid two-phase flow pattern signals based on the multi-scale normalized Benford probability distribution (MSNBPD) analysis. The first focus area is the difference between the characteristics of a multi-scale approach to Benford’s probability distribution and Benford’s Law (BL) distribution in a typical nonlinear system. Once the terms are defined, the authors show how BL could be applied in a multi-scale based flow-pattern based method to analyze differential pressure signals of gas-liquid two-phase flow in small channels to reveal the underlying dynamic characteristics of different gas-liquid two-phase flow patterns, and their sensitivity to scale changes. In addition, the stability of different flow patterns was revealed by the variable scale range calculation. Flow patterns could then be distinguished based on Euclidean distance and the variable scale range difference of differential pressure signals. This paper also compares the present method with other methods that have been used in flow identification. The results show that this new method can well distinguish nonlinear systems with different dynamic mechanisms. This method can also reveal the underlying dynamic characteristics of a gas-liquid two-phase flow of micro and small channels. The two characteristic values (Euclidean distance and variable scale range difference) proved to be a good distinction for two-phase flow patterns than other time and frequency methods, with several possible universal applications. However, the identification results of micro and small channels were better than that of regular pipe, which shows that this method is more suitable for the identification of micro and small channel flow.

The development of bankruptcy prediction models in modern Russian economy

The paper focuses on problems of bankruptcy prediction models in Russia. This issue gains more and more relevance in recent years, in the light of falling incomes of population. The bankruptcy prediction models are widely used to predict bankruptcy. However, they might have low accuracy. In that regard, there is an active discussion in the foreign literature. In Russia there is a growing number of models developed, however so far no detailed studies to assess their efficiency. There are several objectives of the current study: a detailed review of domestic literature to reveal the flaws of existing models; design of a new set of models accounting to analyzed flaws; suggestions for future research. The estimation of Russian models revealed their inefficiency, due to data instability problem, poor financial reports quality, insufficient sample size, and distortion effect of “black accounting” and criminal bankruptcy practices. The set of models offered was proved to be efficient and robust. As the way of further model improvement it was suggested to distinguish between possible scenarios of failure (including liquidation, merger and “freezing”) using statistical classification methods. The Benfords Law analysis was introduced to separate companies falsifying financial statements. The application showed this method as promising.

The “first digit law” – A hypothesis on its possible impact on medicine and development aid

The “first digit law” or “Benford’s law” is a mathematical distribution discovered by Simon Newcomb and Frank Benford. It states, that the probability of the leading number d (d∈{1,…,9}) in many natural datasets follows: P (d)=log10 (d+1)−log10 (d)=log10 (1+1/d).It was successfully used through tax authorities and “forensic accounting” in order to detect fraud and other irregularities. Benfords law was almost neglected for its use outside financial accounting. The planning for health care systems in developing countries is extremely dependant on good, valid data. Whether you plan the catchment area for the future district hospitals, the number of health posts, the staff establishment for the central hospital or the drug budget in the Ministry. The “first digit law” can be used in medicine, public health, physiology and development aid to unmask questionable data, to discover unexpected challenges, difficulties in the data collection process, loss through corruption and criminal fraud. Our hypothesis suggests, that the “first digit law” is a cost effective tool, which is easy to use for most people in the medical profession, which does not really needs complicated statistical software and can be used on the spot, even in the resource restricted conditions of developing countries. Several preconditions (like the size of the data set and its reach over more than two dimensions) have to be fulfilled, but then Benfords law can be used by any clinician, physiologist, public health specialist or aid consultant without difficulties and without deeper statistical knowledge in the four steps, we suggest in this article. The consequences will be different depending on the level (local regional, national, continental, international) on which you will use the law. All levels will be enabled to get insight into the validity of the data-challenges for the other levels without the help of trained statisticians or accountants. We believe that the “first digit law” is a vastly underestimated and neglected, but extremely useful tool for the identification of unexpected challenges, supervision and control in various parts of medicine and public health for almost all aspects of development aid.

Out-phased decadal precipitation regime shift in China and the United States

In order to understand the changes in precipitation variability associated with the climate shift around mid-1970s, the precipitation regime changes have been analyzed over both China and the USA. Specifically, a new variable is designed based on Benford’s Law (BL) to detect precipitation regime shift by using only the first digit information of the datasets. This new variable describes the decadal precipitation regime shift more directly and clearly than the traditional variables, such mean or trend of yearly precipitation amount. It is found that there is an obvious out-phased relation for precipitation regime shift over China and the USA, i.e., a significant shift from the lower to the higher BL’s goodness of fit (weaker to stronger precipitation intensity) in the Southern China occurred in 1979, and a significant shift from the higher to the lower BL’s goodness of fit (stronger to weaker precipitation intensity) in the USA occurred around 1978.

A complete literature review on financial fraud detection applying data mining techniques

Financial fraud is defined as unlawful or criminal duplicity attempted to result to organisational or personal gain. It is a big threat to the economics of a firm, corporate sector, government or ordinary customers in the form of credit card fraud, insurance fraud, and financial statement fraud. Several approaches exist in the literature of financial fraud detection. But, due to the inefficiency of those approaches, researchers leverage data mining techniques to detect financial fraud. This paper aims to build a systematic academic review of financial fraud detection approaches based on data mining techniques in the recent years. In the practice, different data mining techniques namely: K-nearest neighbour, decision tree, fuzzy logic, logistic model, Bayesian belief network, naïve Bayes, Beneish M-Score model, Benfords law, Altman Z-score have been applied to improve the accuracy of fraud detection. In this paper, existing financial fraud detection techniques are compared with the advantages and limitations of the techniques.

Отклонения от закона Бенфорда и распознавание авторских особенностей в текстах

Отклонения от закона Бенфорда и распознавание авторских особенностей в текстах

On the Ability of the Benford's Law to Detect Earthquakes and Discriminate Seismic Signals

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Detecting Cosmetic Debt Management Using Benfords Law

Benfords law states that the frequency of first significant digit in certain samples decreases as those digits increase. This law is used in accounting to find rounding behavior. Several studies provided evidence that firms may round up earnings when they are just below reference points denoted by Nx10k. Most studies are focused on earnings variables. Few studies are focused on other accounting variables like sales for example (Jordan & Alii, 2009; Geyer, 2012). No previous study examines accounting variable from balance sheet (excepted earnings of course). The aim of this short paper is to investigate rounding behavior of long term debt. Using a sample of US public companies, we observe that US firms round down the total long term debt considering two cognitive points: Nx10k and (2xN+1)x5x10j (N is an integer between 1 to 9; k is an integer from 1 and j is an integer from 0). In other words, US public firms exercise Cosmetic Debt Management.

Using Benfords Law To Detect Fraud In The Insurance Industry

Benford's Law is the mathematical phenomena that states that the first digits or left most digits in a list of numbers will occur with an expected logarithmic frequency. While this method has been used in industries such as oil and gas and manufacturing to identify fraudulent activity, it has not been applied to the health insurance industry. Since health insurance companies process a large number of claims each year and these claims are susceptible to fraud, the use of this method in this industry is appropriate. This paper examines the application of Benford's Law to four health insurance companies located in the Midwest. For each company, analysis was performed on the first digit distribution, the first two-digit distribution, and providers with high volumes of claims. The results show that the populations are similar to the frequencies predicted by Benfords Law. The findings also suggested possible fraudulent activity by specific providers, however, the com-panies determined that these results occurred due to abnormal billing practices and were not frau-dulent. The insurance companies that participated in this study will continue to use this method to further detect fraudulent claims.