Benford's Law Research Articles

What Deters Earnings Management? A Benford's Law Comparison Between Publicized Detection Methods and Increased Regulations

ABSTRACTThis paper examines the impact of public awareness and regulatory changes on earnings management over time. Despite scrutiny from regulators, auditors, and investors, the deterrents of earnings management remain unclear. I explore whether public awareness, through the dissemination of research, can influence market behaviors and reduce earnings management activities due to increased detection and penalties. Alternatively, I investigate if stricter regulation plays a role in curbing earnings manipulation. Using Benford's Law, I apply Thomas’ (1989) approach and compare data from 1950 to 1985, 1990 to 2002, and 2003 to 2020. My findings indicate that while public awareness alone does not deter earnings management, heightened regulation, particularly post‐Sarbanes‐Oxley Act (SOX), significantly reduces such practices. This study contributes to the literature by highlighting the comparative impact of public awareness and regulatory changes on earnings management.

Using Benford's Law to Detect Possible Biases in Reported Catches of Tropical Tuna From the Indian Ocean

ABSTRACTAccuracy of catch landing data reported by captains of commercial vessels is crucial in the assessment of marine species stocks and in fisheries policy and management. However, this data can be subject to estimation biases, such as a tendency to inflate some catches (to the detriment of others) and refusal to fill in logbooks. We assessed the accuracy of catch reports from the Spanish tropical tuna purse seine fleet (which accounts for 26% of catches in the Indian Ocean) using Benford's law, a mathematical principle effective for detecting irregularities across multiple datasets. During 2013–2020, including periods before and after the implementation of total allowable catch (TAC) limits in 2017, reported catches differed from Benford's expected distribution, especially for bigeye tuna, indicating potential inaccuracies in reported catches. Changes in data reporting after TAC limits were imposed in 2017 suggested modifications in fishing operations and reporting practices. Use of Benford's law highlighted its potential as an auditing tool in fisheries management and provided insights into data integrity that are crucial for sustainable fisheries governance.

The inefficient effects of non-clinical factors on health care costs.

We use Benford's law to examine the non-random elements of health care costs. We find that as health care expenditures increase, the conformity to the expected distribution of naturally occurring numbers worsens, indicating a tendency towards inefficient treatment. Government insurers follow Benford's law better than private insurers indicating more efficient treatment. Surprisingly, self-insured patients suffer the most from non-clinical cost factors. We suggest that cost saving efforts to reduce non-clinical expenses should be focused on more severe, costly encounters. Doing so focuses cost reduction efforts on less than 10% of encounters that constitute over 70% of dollars spent on health care treatment.

Benford's Law as Debris Flow Detector in Seismic Signals

Benford's Law as Debris Flow Detector in Seismic Signals

Assessing the accuracy of self-reported health expenditure data: Evidence from two public surveys in China

This paper utilizes Benford's law, the distribution that the first significant digit of numbers in certain datasets should follow, to assess the accuracy of self-reported health expenditure data known for measurement errors. We provide both simulation and real data evidence supporting the validity assumption that genuine health expenditure data conform to Benford's law. We then conduct a Benford analysis of health expenditure variables from two widely utilized public datasets, the China Health and Nutrition Survey and the China Family Panel Studies. Our findings show that health expenditure data in both datasets exhibit inconsistencies with Benford's law, with the former dataset tending to be less prone to reporting errors. These results remain robust while accounting for variations in survey design, recall periods, and sample sizes. Moreover, we demonstrate that data accuracy improves with a shorter time interval between hospitalization and interviews, when the data is self-reported as opposed to proxy responses, and at the household level. We find no compelling evidence that enumerators' assessments of respondents' credibility or urgency to end interviews are indicative of data accuracy. This paper contributes to literature by introducing an easy-to-implement analytical framework for scrutinizing and comparing the reporting accuracy of health expenditure data.

Signal processing analysis for detection of anomalies in numerical series

It might be instinctively assumed that the occurrence of the first digit of a randomly selected number is uniformly distributed among 1 to 9. However, the Newcomb–Benford law (NBL), also known as the first-digit law or Benford’s law, reveals a completely different fact: the distributions follows a logarithm-type law. This bias in the frequency of occurrence of the leading digits of numbers is evident in everything from the lengths of rivers, the areas of countries, stock prices, and population numbers. Omnipresence of NBL emerges in mathematics (the Fibonacci sequence, the distribution of prime numbers, etc.) and also in the field of physics (i.e. the half-lives of unstable nuclei). From the earliest years, a large bibliography has been developed and applied to many fields such as fraud detection in finances, international trade and election results analysis. Usually, the diagnosis of such works is reduced to a qualitative binary assessment (conformity or non-conformity). The present paper aims to establish a supervised methodology to address existing gaps and enhance the reliability of final decisions. These gaps include uncertainty about the minimum data volume required to validate Benford’s Law, the low sensitivity of current metrics (Euclidean distance or Mean Absolute Difference) to detect minor levels of adulteration, and the need to quantify both the level and nature of adulteration. Two new procedures have been established coming from Signal Analysis and Processing theory. The first relates the deconvolution between the empirical leading digit distribution and the Benford one. The second procedure consists of obtaining the generating functions (via Hilbert and Fourier transforms) to increase the contrast between both functions: the empirical under research and Benford’s as the reference. Based on calibrated datasets, we are able to quantify the magnitude of accidental errors or malicious fraud adulteration and make the discrimination between both. Finally, as benchmark, fifteen real-world datasets were studied. This study shows that most of datasets are suspicious of Fraud. Notably, in the field of human health, several datasets demonstrate significant fraudulent deviations, defying conventional patterns of fraud and error. This phenomenon has been termed Berserker Fraud.

Measuring Bedload Motion Time at Second Resolution Using Benford's Law on Acoustic Data

AbstractBedload transport is a natural process that strongly affects the Earth's surface system. An important component of quantifying bedload transport flux and establishing early warning systems is the identification of the onset of bedload motion. Bedload transport can be monitored with high temporal resolution using passive acoustic methods, for example, hydrophones. Yet, an efficient method for identifying the onset of bedload transport from long‐term continuous acoustic data is still lacking. Benford's Law defines a probability distribution of the first‐digit of data sets and has been used to identify anomalies. Here, we apply Benford's law to continuous acoustic recordings from Baiyang hydrometric station, a tributary of Liwu River, Taroko National Park, Taiwan at the frequency of 32 kHz from stationary hydrophones deployed for 3 years since 2019. We construct a workflow to parse sound combinations of bedload transportation and analyze them in the context of hydrometric sensing constraining the onset, and recession of bedload transport. We identified three separate sound classes in the data related to the noise produced by the motion of pebbles, water flow, and air. We identify two bedload transport events that lasted 17 and 45 hr, respectively, covering about 0.35% of the total recorded time. The workflow could be transferred to other different catchments, events, or data sets. Due to the influence of instrument and background noise on the regularity of the residuals of the first‐digit, we recommend identifying the first‐digit distribution of the background noise and ruling it out before implementing this workflow.

Применение статистик m* и d* для оценки качества данных психологического исследования с помощью закона Бенфорда (на примере измерения времени реакции)

<p>The objective of the proposed study was to examine the properties of statistics used to assess the conformity of the distribution of the first significant digit to Benford's Law, m* and d*, with relatively modest sample sizes (10≤n ≤70). A simulation study was conducted to achieve this goal. Data were simulated following a log-normal distribution with parameters that mimic the distribution of reaction time measurements. The distribution of the first significant digit was examined in standardized values raised to the power of γ; 5≤γ≤100. It was found that the statistic m* does not depend on the power of the number, unlike d*. Critical values were established for samples ranging from 10 to 70 observations with an increment of h=10. It turned out that for small n, the critical values of the statistic d* are close to asymptotic, while the critical values of the statistic m* are significantly larger. The functionality of the established critical values was tested within the framework of an experimental study: one respondent performed the Stroop cognitive test in accordance with the instructions (control case), while another violated them (experimental case). It was discovered that the statistic d* does not allow for differentiation in the behavior of subjects. Conversely, m* proved sensitive to changes in respondent behavior, and in the experimental case, it significantly more often allowed for the rejection of the null hypothesis regarding the conformity of the distribution of the first significant digit of the standardized value of reaction time to Benford's Law compared to the control. Thus, a preliminary conclusion is made that the statistic m* is more functional compared to d* in studying the quality of data on reaction time with small n.</p>

Benford's law and random integer decomposition with congruence stopping condition

Benford's law is a statement about the frequency that each digit arises as the leading digit of numbers in a dataset. It is satisfied by various common integer sequences, such as the Fibonacci numbers, the factorials, and the powers of most integers. In this paper, we prove that integer sequences resulting from a random integral decomposition process (which we model as discrete “stick breaking”) subject to a certain congruence stopping condition approach Benford distribution asymptotically. We also show that our requirement on the number of congruence classes defining the congruence stopping condition is necessary for Benford behavior to occur and is a critical point; deviation from that would result in drastically different behavior.

Validating self-reported Toxic Release Inventory data using Benford's Law: investigating toxic chemical release hazards in floodplains.

Acute and long-term health impacts from flooding related toxic chemical releases are a significant local health concern and can disproportionately impact communities with vulnerable populations; reliable release data are needed to quantify this hazard. In this paper, we analyze US Federal Emergency Management Agency designated floodplain data and US Environmental Protection Agency Toxic Release Inventory (TRI) data to determine if geographically manipulated databases adhere to Benford's Law. We investigated multiple variants and discovered pollution releases adhere to Benford's Law and tests which thereby validates the self-reported toxic release dataset. We find that Benford's Law applies to self-reported toxic chemical release and disposal data, indicating a lack of widespread data errors or manipulation.

Potential COVID-19 test fraud detection: Findings from a pilot study comparing conventional and statistical approaches.

Some COVID-19 testing centres have reported manipulated test numbers for antigen tests/rapid tests. This study compares statistical approaches with traditional fraud detection methods. The extent of agreement between traditional and statistical methods was analysed, as well as the extent to which statistical approaches can identify additional cases of potential fraud. Outlier detection marking a high number of tests, modeling of the positivity rate (Poisson Regression), deviation from distributional assumptions regarding the first digit (Benford's Law) and the last digit of the number of reported tests. The basis of the analyses were billing data (April 2021 to August 2022) from 907 testing centres in a German city. The positive agreement between the conventional and statistical approaches ('sensitivity') was between 8.6% and 24.7%, the negative agreement ('specificity') was between 91.3% and 94.6%. The proportion of potentially fraudulent testing centres additionally identified by statistical approaches was between 7.0% and 8.7%. The combination of at least two statistical methods resulted in an optimal detection rate of test centres with previously undetected initial suspicion. The statistical approaches were more effective and systematic in identifying potentially fraudulent testing centres than the conventional methods. Testing centres should be urged to map paradata (e.g. timestamps of testing) in future pandemics.

Application of citizen science with the nationwide bird census

Citizen science (CS) is the most effective tool for overcoming the limitations of government and/or professional data collection. To compensate for quantitative limitations of the 'Winter Waterbird Census of Korea', we conducted a total of four bird monitoring via CS from 2021 to 2022. To use CS data alongside national data, we studied CS data quality and improvement utilizing (1) digit-based analysis using Benford’s law and (2) comparative analysis with national data. In addition, we performed bird community analysis using CS-specific data, demonstrating the necessity of CS. Neither CS nor the national data adhered to Benford's law. Alpha diversity (number of species and Shannon index) was lower, and total beta diversity was higher for the CS data than national data. Regarding the observed bird community, the number of species per family was similar; however, the number of individuals per family/species differed. We also identified the necessity of CS by confirming the possibility of predicting bird communities using CS-specific data. CS was influenced by various factors, including the perceptions of the survey participants and their level of experience. Therefore, conducting CS after systematic training can facilitate the collection of higher-quality data

Comparative Analysis of Dynamic Power Consumption of Parallel Prefix Adder

The Newcomb-Benford law, also known as Benford's law, is the law of anomalous numbers stating that in many real-life numerical datasets, including physical and statistical ones, numbers have a small initial digit. Numbers irregularity observed in nature leads to the question, is the arithmetical-logical unit, responsible for performing calculations in computers, optimal? Are there other architectures, not as regular as commonly used Parallel Prefix Adders, that can perform better, especially when operating on the datasets that are not purely random, but irregular? In this article, structures of a propagate-generate tree are compared including regular and irregular configurations—various structures are examined: regular, irregular, with gray cells only, with both gray and black, and with higher valency cells. Performance is evaluated in terms of energy consumption. The evaluation was performed using the extended power model of static CMOS gates. The model is based on changes of vectors, naturally taking into account spatio-temporal correlations. The energy parameters of the designed cells were calculated on the basis of electrical (Spice) simulation. Designs and simulations were done in the Cadence environment and calculations of the power dissipation were performed in MATLAB. The results clearly show that there are PPA structures that perform much better for a specific type of numerical data. Negligent design can lead to an increase greater than two times of power consumption. The novel architectures of PPA described in this work might find practical applications in specialized adders dealing with numerical datasets, such as, for example, sine functions commonly used in digital signal processing.

Do ESG funds engage in portfolio pumping to gain higher flows? An application of Benford's Law

AbstractPortfolio pumping is identified as an ‘illegal’ trading practice that involves inflating quarter‐ and year‐end portfolio returns. Utilizing U.S. domestic equity mutual fund daily return data, we examine whether environmental, social, and governance (ESG) funds engage in portfolio pumping to generate higher flows. Our findings reveal that, on average, ESG funds attract 0.4% higher flows than other funds. However, they engage in portfolio pumping and achieve returns that are 5.3 basis points (bps) higher at quarter ends compared to their returns during the rest of the year. This practice does not result in additional fund flows. Notably, compared to other funds, ESG funds exhibit a significant 4 bps reduction through portfolio pumping. This implies that, on the last day of the quarter, ESG funds earn approximately 4 bps lower returns compared to other funds. Portfolio pumping is costly for both investors and financial markets since it leads to trading activities that cause stock prices to deviate from their fundamental values. ESG funds engage less in portfolio pumping than other funds, which indicates that their primary focus is to maximize fund flows rather than enhance or create a positive social impact on the underlying firm portfolio. Investors seem to understand this goal, particularly when ESG funds engage in portfolio pumping and avoid investing in ESG funds.

A novel approach to evaluate data integrity: evidence from COVID-19 in China

The COVID-19 pandemic has generated an unprecedented amount of epidemiological data. Yet, concerns regarding the validity and reliability of the information reported by health surveillance systems have emerged worldwide. In this paper, we develop a novel approach to evaluating data integrity by combining the Newcomb-Benford Law with outlier methods. We demonstrate the advantages of our framework using a case study from China. To ensure more robust findings, we employ multiple diagnostic procedures, including three conformity estimates, four goodness-of-fit tests, and two distance measures (Cook and Mahalanobis). To promote transparency, we have made all computational scripts publicly available. Our findings indicates a significant deviation in the distribution of new deaths from the theoretical expectations of Benford's Law. Importantly, these results remain accurate even when considering alternative model specifications and conducting various statistical tests. Furthermore, the procedures developed here are easily applicable in other areas of knowledge and can be scaled to assess data quality in both the public and private sectors.

Benford’s law analysis to evaluate the quality data of COVID-19 epidemiological surveillance in Indonesia

Countries worldwide, including Indonesia, grappled with the unprecedented challenges brought about by the coronavirus disease (COVID-19) pandemic. Surveillance data vividly illustrates the profound effect of the COVID-19 pandemic in Indonesia. Both daily cases and deaths were raised, revealing the rapid transmission of the virus within communities. A quantitative study using a statistical approach was accomplished with secondary data to evaluate the quality of COVID-19 epidemiological surveillance data in Indonesia during the period between March 2020 to January 2021. The data was sourced from the World Health Organization (WHO) website using data reports on COVID-19 confirmed cases and deaths. A rapid tool called the first digit law or the fulfillment of Benford’s law was used to suggest good data quality for epidemiological surveillance. Data analysis used the Chi-squared test and the log-likelihood ratio test. Also, it displayed the difference in mean absolute deviation (MAD) to identify the proximity of the data and Benford’s law distribution. The results showed that both confirmed, and death case distributions were statistically non-conformity with Benford’s law distribution. In terms of quality data regarding the COVID-19 pandemic, the epidemiological surveillance system falls short of Benford’s law assumption. Benford's law has been acknowledged as an initial analysis that can expeditiously assess the performance of a surveillance system. The next phase of this study would be to conduct a complete evaluation suitably, especially in post-pandemic COVID-19.

Some new invariant sum tests and MAD tests for the assessment of Benford’s law

AbstractThe Benford law is used world-wide for detecting non-conformance or data fraud of numerical data. It says that the significand of a data set from the universe is not uniformly, but logarithmically distributed. Especially, the first non-zero digit is One with an approximate probability of 0.3. There are several tests available for testing Benford, the best known are Pearson’s $$\chi ^2$$ χ 2 -test, the Kolmogorov–Smirnov test and a modified version of the MAD-test. In the present paper we propose some tests, three of the four invariant sum tests are new and they are motivated by the sum invariance property of the Benford law. Two distance measures are investigated, Euclidean and Mahalanobis distance of the standardized sums to the orign. We use the significands corresponding to the first significant digit as well as the second significant digit, respectively. Moreover, we suggest inproved versions of the MAD-test and obtain critical values that are independent of the sample sizes. For illustration the tests are applied to specifically selected data sets where prior knowledge is available about being or not being Benford. Furthermore we discuss the role of truncation of distributions.

Benford's law: Planning and analysis of the planned values in the defense budget

Benford's law: Planning and analysis of the planned values in the defense budget

지방자치단체 재무회계정보의 신뢰성이 나아지고 있는가?

It has been 16 years since financial accounting for local governments using accrual based double-entry bookkeeping was introduced. The purpose of this study is to verify whether the reliability of financial statement information of local governments is improving by using Benford's Law. The reliability of the numbers was estimated by comparing the distribution of the first digit of the assets and liabilities with the Benford distribution. As a result of the analysis, it is confirmed that the assets and liabilities for the past 16 years did not follow the Benford distribution as the MAD test result value and the KS statistic exceeded the critical value. Assets and assets in each statement of financial position for the first 8 years of the introduction of the accrual-based double-entry bookkeeping system and for the first 8 years after financial statements began to be included as mandatory documents in the settlement of accounts of local governments. Liabilities figures still does not follow the Benford distribution. We suggest that there is a need to move from a simple review system to an audit system for local governments' financial statements.

This Unexpected Pattern of Numbers Is Everywhere: A curious mathematical phenomenon called Benford's law governs the numbers all around us.

This Unexpected Pattern of Numbers Is Everywhere: A curious mathematical phenomenon called Benford's law governs the numbers all around us.