# Is Benford’s Law a Universal Behavioral Theory?

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## Abstract

**:**

## 1. Introduction

## 2. A Behavioral System Example

#### 2.1. Australia’s Income Distribution

#### 2.2. FSD Distributions for Australia

**Figure 3.**The distribution of Australian income FSDs for the years 2008–2013 and Benford Distribution.

**Table 1.**Chi-square and correlation between Benford’s and income first significant digit (FSD), 2008–2013.

Statistic | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 |
---|---|---|---|---|---|---|

Chi-square | 0.031 | 0.046 | 0.031 | 0.029 | 0.026 | 0.017 |

Correlation | 0.965 | 0.953 | 0.968 | 0.971 | 0.975 | 0.987 |

Significance | 0.999 | 0.999 | 0.999 | 0.9999 | 0.9999 | 0.9999 |

## 3. Entropy Based Estimation and Inference

#### 3.1. Problem Formulation and Solution

#### 3.2. An Information Theoretic Approach

#### 3.3. CR ($\gamma \to 1$) Mean Related FSD Distribution

**Table 2.**Estimated empirical likelihood (EL) distributions (with uniform reference distribution) for the FSD problem and their correlation (r) with Benford’s distribution.

FSD Mean | ${\widehat{p}}_{1}$ | ${\widehat{p}}_{2}$ | ${\widehat{p}}_{3}$ | ${\widehat{p}}_{4}$ | ${\widehat{p}}_{5}$ | ${\widehat{p}}_{6}$ | ${\widehat{p}}_{7}$ | ${\widehat{p}}_{8}$ | ${\widehat{p}}_{9}$ | r |
---|---|---|---|---|---|---|---|---|---|---|

3.0 | 0.395 | 0.173 | 0.111 | 0.082 | 0.065 | 0.053 | 0.046 | 0.040 | 0.035 | 0.990 |

3.44 | 0.300 | 0.177 | 0.125 | 0.097 | 0.079 | 0.067 | 0.058 | 0.051 | 0.046 | 1.000 |

4.0 | 0.208 | 0.161 | 0.132 | 0.111 | 0.096 | 0.085 | 0.076 | 0.068 | 0.062 | 0.980 |

#### 3.4. Discussion

## 4. An Information Theoretic Example

**Table 3.**Chi Square values, correlations and significance for Australian yearly empirical likelihood (EL) distributions and the Benford FSD distributions.

Statistic | 2008EL | 2009EL | 2010EL | 2011EL | 2012EL | 2013EL |
---|---|---|---|---|---|---|

Chi square | 0.0016 | 0.0000 | 0.0001 | 0.0002 | 0.0009 | 0.0015 |

Correlation | 0.9994 | 1.0000 | 1.0000 | 1.0000 | 0.9998 | 0.9997 |

Significance | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix

#### Maximum Entropy Distribution

**Figure A1.**Australian Maximum Entropy Distribution and Benford FSD Distribution for the aggregate years 2008–2013. Chi square: 0.026945346, Correlation: 0.962189049.

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^{2}Among many who have used Benford’s law to check the validity of purported scientific data in the social sciences see [1,2] for a survey. For instance, the law has been used in census data, eBay auction prices, users of on-line social networks, voting fraud, macroeconomic data, and religious activity.^{3}The data can be obtained from the webpage http://www.melbourneinstitute.com/hilda/data/ by submitting a request for access.

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**MDPI and ACS Style**

Villas-Boas, S.B.; Fu, Q.; Judge, G.
Is Benford’s Law a Universal Behavioral Theory? *Econometrics* **2015**, *3*, 698-708.
https://doi.org/10.3390/econometrics3040698

**AMA Style**

Villas-Boas SB, Fu Q, Judge G.
Is Benford’s Law a Universal Behavioral Theory? *Econometrics*. 2015; 3(4):698-708.
https://doi.org/10.3390/econometrics3040698

**Chicago/Turabian Style**

Villas-Boas, Sofia B., Qiuzi Fu, and George Judge.
2015. "Is Benford’s Law a Universal Behavioral Theory?" *Econometrics* 3, no. 4: 698-708.
https://doi.org/10.3390/econometrics3040698