Equipartitions and a distribution for numbers: A statistical model for Benford's law

Iafrate, Joseph R.; Miller, Steven J.; Strauch, Frederick W.

doi:10.1103/physreve.91.062138

Cited by 12 publications

(6 citation statements)

References 24 publications

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“…As already stated, one of the main findings of our research has been about the data range. Indeed to conform with the law, the data set must contain data in which each number 1 through 9 has an equal chance of being the leading digit; there should be equipartition [42,43]. However, this seems paradoxical.…”

Section: Discussionmentioning

confidence: 99%

Benford’s laws tests on S&P500 daily closing values and the corresponding daily log-returns both point to huge non-conformity

Ausloos

Ficcadenti

Dhesi

et al. 2021

Physica A: Statistical Mechanics and its Applications

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The so-called Benford's laws are of frequent use to detect anomalies and regularities in data sets, particularly in election results and financial statements. However, primary financial market indices have not been much studied, if studied at all, within such a perspective.This paper presents features in the distributions of S&P500 daily closing values and the corresponding daily log-returns over a long time interval, [03/01/1950 -22/08/2014], amounting to 16265 data points. We address the frequencies of the first, second, and first two significant digits and explore the conformance to Benford's laws of these distributions at five different (equal size) levels of disaggregation. The log-returns are studied for either positive or negative cases. The results for the S&P500 daily closing values are showing a remarkable lack of conformity, whatever the different levels of disaggregation. The causes of this non-conformity are discussed, pointing to the danger in taking Benford's laws for granted in large databases, whence drawing ''definite conclusions''. The agreements with Benford's laws are much better for the log-returns. Such a disparity in agreements finds an explanation in the data set itself: the index's inherent trends. To further validate this, daily returns have been simulated via the Geometric Brownian Motion and calibrating the simulations with the observed data averages and testing against Benford's laws when the log-returns distribution's standard deviation changes. One finds that the trend and the standard deviation of the distributions are relevant parameters in concluding about conformity with Benford's laws.

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Section: Discussionmentioning

confidence: 99%

Benford’s laws tests on S&P500 daily closing values and the corresponding daily log-returns both point to huge non-conformity

Ausloos

Ficcadenti

Dhesi

et al. 2021

Physica A: Statistical Mechanics and its Applications

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“…If this partition is done so as to maximize the entropy of the partition, we find that the probability of city size t is given by (2), where λ = M/P. See Appendix A for a derivation of this claim that is inspired by a similar derivation by Iafrate, Miller, and Strach [9]. Miller and Nigrini [10] also explore relations between the exponential probability density (2) and Benford's law (1).…”

Section: Introductionmentioning

confidence: 95%

First Digit Oscillations

Lemons

Peter

2021

Stats

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“…Although this is an expected statistical phenomenon [3], the reasons for it are remarkably obscure, and there remain a number of open problems [4]. Benford's Law has since generated significant interest, including treatments that highlight its connections with entropy: Iafrate et al (2015) [5] showed that the Law is derivable from a statistical mechanics treatment, with Don Lemons (2019) [6] extending their treatment to explicitly show the connection with thermodynamics.…”

Section: Introductionmentioning

confidence: 99%

A Maximum Entropy Resolution to the Wine/Water Paradox

Parker,

Jeynes

2023

Entropy

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The Principle of Indifference (‘PI’: the simplest non-informative prior in Bayesian probability) has been shown to lead to paradoxes since Bertrand (1889). Von Mises (1928) introduced the ‘Wine/Water Paradox’ as a resonant example of a ‘Bertrand paradox’, which has been presented as demonstrating that the PI must be rejected. We now resolve these paradoxes using a Maximum Entropy (MaxEnt) treatment of the PI that also includes information provided by Benford’s ‘Law of Anomalous Numbers’ (1938). We show that the PI should be understood to represent a family of informationally identical MaxEnt solutions, each solution being identified with its own explicitly justified boundary condition. In particular, our solution to the Wine/Water Paradox exploits Benford’s Law to construct a non-uniform distribution representing the universal constraint of scale invariance, which is a physical consequence of the Second Law of Thermodynamics.

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Equipartitions and a distribution for numbers: A statistical model for Benford's law

Cited by 12 publications

References 24 publications

Benford’s laws tests on S&P500 daily closing values and the corresponding daily log-returns both point to huge non-conformity

Benford’s laws tests on S&P500 daily closing values and the corresponding daily log-returns both point to huge non-conformity

First Digit Oscillations

A Maximum Entropy Resolution to the Wine/Water Paradox

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