scholarly journals A local Benford Law for a class of arithmetic sequences

2019 ◽  
Vol 15 (03) ◽  
pp. 613-638
Author(s):  
Zhaodong Cai ◽  
A. J. Hildebrand ◽  
Junxian Li
Keyword(s):  
Large Class ◽  
Local Level ◽  
Fibonacci Numbers ◽  
Point Of View ◽  
Benford’S Law ◽  
Local Distribution ◽  
Local Point ◽  
Arithmetic Sequences ◽  
Benford's Law ◽  
Benford Law

It is well known that sequences such as the Fibonacci numbers and the factorials satisfy Benford’s Law; that is, leading digits in these sequences occur with frequencies given by [Formula: see text], [Formula: see text]. In this paper, we investigate leading digit distributions of arithmetic sequences from a local point of view. We call a sequence locally Benford distributed of order [Formula: see text] if, roughly speaking, [Formula: see text]-tuples of consecutive leading digits behave like [Formula: see text] independent Benford-distributed digits. This notion refines that of a Benford distributed sequence, and it provides a way to quantify the extent to which the Benford distribution persists at the local level. Surprisingly, most sequences known to satisfy Benford’s Law have rather poor local distribution properties. In our main result we establish, for a large class of arithmetic sequences, a “best-possible” local Benford Law; that is, we determine the maximal value [Formula: see text] such that the sequence is locally Benford distributed of order [Formula: see text]. The result applies, in particular, to sequences of the form [Formula: see text], [Formula: see text], and [Formula: see text], as well as the sequence of factorials [Formula: see text] and similar iterated product sequences.

Using Benford’s law on the seismic reflectivity analysis

2018 ◽  
Vol 6 (3) ◽  
pp. T689-T697
Author(s):  
Isadora A. S. de Macedo ◽  
Jose Jadsom S. de Figueiredo
Keyword(s):  
Geomagnetic Field ◽  
Point Of View ◽  
P Wave ◽  
Benford’S Law ◽  
Common Procedure ◽  
Data Set ◽  
Benford's Law ◽  
Distribution Point ◽  
Seismic Reflectivity ◽  
Reflectivity Data

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.

Introduction

2015 ◽  
Author(s):  
Arno Berger ◽  
Theodore P. Hill
Keyword(s):  
Numerical Data ◽  
Benford’S Law ◽  
Decimal Digit ◽  
Mathematical Framework ◽  
Benford's Law ◽  
History Of ◽  
Logarithmic Distribution ◽  
Benford Law ◽  
Special Case

This introductory chapter provides an overview of Benford' law. Benford's law, also known as the First-digit or Significant-digit law, is the empirical gem of statistical folklore that in many naturally occurring tables of numerical data, the significant digits are not uniformly distributed as might be expected, but instead follow a particular logarithmic distribution. In its most common formulation, the special case of the first significant (i.e., first non-zero) decimal digit, Benford's law asserts that the leading digit is not equally likely to be any one of the nine possible digits 1, 2, … , 9, but is 1 more than 30 percent of the time, and is 9 less than 5 percent of the time, with the probabilities decreasing monotonically in between. The remainder of the chapter covers the history of Benford' law, empirical evidence, early explanations and mathematical framework of Benford' law.

Benford Law and Earnings Analysis

2020 ◽  
pp. 381-404
Author(s):  
Radiah Othman ◽  
Rashid Ameer
Keyword(s):  
Corporate Governance ◽  
Rule Of Law ◽  
Institutional Factors ◽  
Benford’S Law ◽  
Government Effectiveness ◽  
The Rule Of Law ◽  
Benford's Law ◽  
Governance Quality ◽  
Corporate Governance Quality ◽  
Benford Law

The literature suggests that firms are actively managing the smoothing of their reported positive net incomes. The observed frequency of second digits abnormally exceeds the level predicted by Benford's Law, which results in a higher frequency of the number zero and an abnormally low occurrence of the number nine in the second digit of the reported income numbers. A reversal pattern occurs for reported net losses. This phenomenon is typically peculiar to countries with weak governance and firms under pressure to meet analysts' expectations. This chapter examines 10 years of reported net incomes by 5,040 firms (44,636 firm-years) in 10 countries ranked as having the best corporate governance quality. The analysis reveals that firms in these countries were not spared from opportunistically rounding their earning numbers. In fact, this rounding behavior is more prevalent when net losses were reported and this rounding phenomenon co-varied with some institutional factors; in particular, the rule of law and government effectiveness has significantly influenced the rounding behavior.

scholarly journals APLIKASI HUKUM BENFORD DALAM MENGANALISA KASUS GARUDA INDONESIA

2020 ◽  
Vol 13 (2) ◽  
Author(s):  
Theresia Hesti Bwarleling
Keyword(s):  
Distribution Pattern ◽  
Financial Statements ◽  
Benford’S Law ◽  
Average Absolute Deviation ◽  
Absolute Deviation ◽  
Audit Planning ◽  
Benford's Law ◽  
Benford Law

<p><strong>ABSTRACT</strong>: Benford's Law is a simple and effective tool in finding fraud for an auditor. The purpose of this study was to determine whether Benford's Law can be used effectively in analyzing the indication of ambiguity in the financial statements of PT. Garuda Indonesia (Persero) Tbk. 2018. To answer this question, this research uses all accounts in the company's financial statements from 2018 and 2017. In assessing whether these accounts have differences in distribution with Benford's Law, this study uses the average absolute deviation. Then the predicted absolute deviation results are tested by referring to the exposure of Garuda's incidental publications (July, 2019). The results of this study indicate that the value of the financial statements of PT. Garuda Indonesia (Persero) Tbk. in the year 2018 fulfills the Benford Law distribution pattern. Meanwhile, in 2017 there were several deviant account values which were then asked to be restated in the 2018 reporting. Benford's law has once again proven to be used as an effective tool in audit planning.</p><p><strong>Keywords:</strong> Benford Law, Garuda Indonesia Financial Statements, Average Absolute Deviation</p><p align="center"><strong> </strong></p><p><strong>ABSTRAK:</strong> Hukum Benford merupakan alat bantu yang sederhana dan efektif dalam menemukan kecurangan bagi seorang auditor. Tujuan dari penelitian ini adalah untuk mengetahui apakah Hukum Benford dapat digunakan secara efektif dalam menganalisa adanya indikasi kerancuan Laporan Keuangan PT. <a href="https://www.cnnindonesia.com/tag/garuda-indonesia">Garuda Indonesia</a> (Persero) Tbk. Tahun 2018. Untuk menjawab pertanyaan tersebut, riset ini menggunakan seluruh akun dalam laporan keuangan perusahaan tahun 2018 dan 2017. Dalam menilai apakah akun-akun tersebut memiliki perbedaan distribusi dengan Hukum Benford, penelitian ini menggunakan rata-rata deviasi absolut. Kemudian hasil rata-rata deviasi absolut diuji prediksinya dengan merujuk pada paparan publikasi insidentil Garuda (Juli, 2019). Hasil dari penelitian ini menunjukkan bahwa nilai akun Laporan Keuangan PT. <a href="https://www.cnnindonesia.com/tag/garuda-indonesia">Garuda Indonesia</a> (Persero) Tbk. Tahun 2018 memenuhi pola distribusi Hukum Benford. Sedangkan pada tahun 2017 terdapat beberapa nilai akun yang menyimpang dan kemudian diminta untuk disajikan ulang di pelaporan tahun 2018. Hukum Benford sekali lagi terbukti dapat digunakan sebagai alat bantu yang efektif dalam perencanaan audit.</p><p><strong>Kata Kunci: </strong>Hukum Benford, Laporan Keuangan <a href="https://www.cnnindonesia.com/tag/garuda-indonesia">Garuda Indonesia</a>, Rata-rata Deviasi Absolut</p>

Benfordʼs Law as a Bridge between Statistics and Accounting

Benford's Law ◽  
2015 ◽  
Author(s):  
Richard J. Cleary ◽  
Jay C. Thibodeau
Keyword(s):  
Statistics Education ◽  
Point Of View ◽  
Accounting Students ◽  
Benford’S Law ◽  
Statistical Point ◽  
Accounting Practice ◽  
Current State ◽  
Key Concepts ◽  
Benford's Law

This chapter explores the connections between statistics and accounting through Benford's law and considers the questions of when and how to effectively deliver this material in such a way that Benford's law can be a tool that helps accountants make stronger and more efficient decisions using sound statistical practice. It looks at the current state of statistics education for accounting students and presents some of the ways in which accounting practice, particularly in auditing, can benefit from a statistical point of view. The chapter then demonstrates how Benford's law can be used to reinforce the key concepts that appear at the intersection of ideas from statistics and accounting. Finally, the chapter concludes with some suggestions for how to effectively incorporate Benford's law into the curriculum.

Fast growing sequences of numbers and the first digit phenomenon

2015 ◽  
Vol 11 (03) ◽  
pp. 705-719 ◽  
Author(s):  
Bruno Massé ◽  
Dominique Schneider
Keyword(s):  
Large Class ◽  
Design Process ◽  
Benford’S Law ◽  
Fast Growing ◽  
Benford's Law ◽  
Natural Density

We consider a large class of fast growing sequences of numbers Un like the nth superfactorial [Formula: see text], the nth hyperfactorial [Formula: see text] and similar ones. We show that their mantissas are distributed following Benford's law in the sense of the natural density. We prove that this is also verified by [Formula: see text], by [Formula: see text] and is passed down to all the sequences obtained by iterating this design process. We also consider the superprimorial numbers and the products of logarithms of integers.

scholarly journals THE USE OF BENFORD’S LAW IN PERFORMANCE AUDIT TO DETECT FRAUD IN THE STATE EXPENDITURETRANSACTIONS ASSISTED BY IDEA SOFTWARE (A Case Study at the Transportation Ministry Work Units)

2020 ◽  
Vol 5 (1) ◽  
pp. 147
Author(s):  
Eka C. Setyawan
Keyword(s):  
Research Method ◽  
Benford’S Law ◽  
Transaction Data ◽  
Benford's Law ◽  
State Expenditure ◽  
Direct Experiment ◽  
Work Units ◽  
The Right ◽  
Benford Law

The aim ofthis study was to determine the use of Benford's Law in Performance Audits todetect fraudin state expenditure transactions assisted by IDEA Software,a case studyat several Work Units of the Ministry of Transportation. The research method used was descriptive qualitative and direct experiment. This researchwas conucted while the researcher wascarruing outperformance auditat three work units of the Ministry of Transportation. The researcher used IDEA Software to analyze the financial transaction data according to Benford's law. Based on the results of the research, the right type of state expenditure to be analyzed using Benford law isgoods expenditure.In addition,Benford’s Law can be used in performance audits to determinestate expenditure transactions that are indicated fraud with the success rate of ranging from 20% to 50%t. Transactions that appear are not in accordance with Benford's law.

Forensic Auditing Tools in Detecting Financial Statements' Irregularities

2019 ◽  
pp. 256-275 ◽  
Author(s):  
Radiah Othman ◽  
Rashid Ameer ◽  
Fawzi Laswad
Keyword(s):  
Balance Sheet ◽  
Financial Statements ◽  
Benford’S Law ◽  
Total Current ◽  
Audit Planning ◽  
Income Statement ◽  
Benford's Law ◽  
Sampling Procedures ◽  
Benford Law

This chapter illustrates a three-stage analytical procedure to examine and detect the likelihood of financial statements manipulation and identify the accounts that were manipulated by Toshiba. It applies the Beneish model and Benford's law to Toshiba's balance sheet and income statement from 2002 to 2016. The results show significant deviation from Benford's law in the pre-fraud period in equity, long-term receivables and property, plant and equipment, long-term liabilities, and in the post-fraud period in the long-term liabilities, equity, long-term receivables, and total current assets. The results provide evidence of the usefulness of Beneish and Benford law as forensic auditing tools for detecting financial statements' irregularities and fraud that would be useful for the audit planning and sampling procedures.

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