Risque associé à l’utilisation de la loi de Benford pour détecter les fraudes dans le secteur de la mode [Risk of Reviews based on Benford Law in the Fashion. Français: Fréquences relatives d’apparition de la 1ère décimale d’un résultat de mesure selon la Loi de Benford Licence: Date, 31 March A Simple Explanation of Benford’s Law. R. M. FEWSTER. Benford’s Law, also known as the first-digit law, has long been seen as a tantalizing and mysterious.

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His data set included the surface areas of rivers, the sizes of US populations, physical constantsmolecular weightsentries from a mathematical handbook, numbers contained in an issue of Reader’s Digestthe street addresses of the first persons listed in American Men of Science and death rates.

The number of open reading frames and their relationship to genome size differs between eukaryotes and prokaryotes with the former showing a log-linear relationship and the latter a linear relationship.

By contrast, that hypothetical stock price described above can be written as the product of many random variables i. While the first digits of nominal prices distributed according to Benford’s Law, the study showed a clear deviation from this benchmark for the second and third digits in nominal market prices with a clear trend towards psychological pricing after the nominal shock of the euro introduction. The reason is that the logarithm of the stock price is undergoing a random walkso over time its probability distribution will get more and more broad and smooth see above.

Alessandro Gambini; et al. InTed Hill proved the result about mixed distributions mentioned below. See, for example, [1]. Tom Van Caneghem, Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher soliton discrete uniform Zipf Zipf—Mandelbrot. The ratio of two log normal distributions is a log normal so this distribution was not examined. RePEc uses bibliographic data supplied by the respective publishers. International Journal of Research in Marketing.

This can also be proven mathematically: These critical values provide the minimum test statistic values required to reject the hypothesis of compliance with Benford’s law lloi the given significance levels.

It is possible to extend the law to digits beyond the first. Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal bneford t matrix gamma normal-inverse-Wishart normal-Wishart Wishart.

Morris, Richard James, ed. You can help correct errors and omissions.

Based on the plausible assumption that people who make up figures tend to distribute their digits fairly uniformly, a simple comparison of first-digit frequency distribution from the data with the expected distribution according to Benford’s Law ought to show up any anomalous results.

Other distributions that have been examined include the Muth distributionGompertz distributionWeibull distributiongamma distributionlog-logistic distribution and the exponential power distribution all of which show reasonable agreement with the law. Statistical Mechanics and its ApplicationsElsevier, vol. If the goal is to conclude agreement with the Benford’s law rather than disagreement, then the goodness-of-fit tests mentioned above are inappropriate.

There is a generalization of the law to numbers expressed in other bases for example, base 16and also a generalization from leading 1 digit to leading n ooi. Like other general principles about natural data — brnford example the fact that many data sets are well approximated by a normal distribution — there are illustrative examples and explanations that cover many of the cases where Benford’s law applies, though there are many other cases where Benford’s law applies that resist a simple explanation.

Neither the right-truncated normal distribution nor the ratio distribution venford two right-truncated normal distributions are well described by Benforc law. As the distribution gets narrower, the discrepancies from Benford’s law typically increase gradually. One is an exponential growth or decay process: Benford’s lawalso called Newcomb-Benford’s lawlaw of anomalous numbersand first-digit lawis bengord observation about the frequency distribution of leading digits in many real-life sets of numerical data.

It has been shown that this result applies to a wide variety of data sets, including electricity bills, street addresses, stock prices, house prices, population numbers, death rates, lengths of rivers, physical and mathematical constants [3].

Nigrini [33] has suggested the use of lou z statistic. If one writes all those lengths in meters, or writes them all in feet, it is reasonable to expect that the distribution of first digits should be the same on the two lists.

## Mathematics > Dynamical Systems

An empirical investigation and a novel explanation”. Many real-world examples of Benford’s law arise from multiplicative fluctuations.

Nicolas Gauvrit; Jean-Paul Delahaye Square roots and reciprocals do not obey this law. Benford’s Law and psychological pricing after the euro introduction”.

Help us Corrections Found an error or omission? Consider the probability distributions shown below, referenced to a log scale. Then, I present the bias and their significance. In the United States, evidence based on Benford’s law has been admitted in criminal cases at the federal, state, and local levels. A narrow probability distribution of the log of a variable, shown on a log scale [11].

### [] Relations de r\’ecurrence lin\’eaires, primitivit\’e et loi de Benford

MathWorld, A Wolfram web resource. Rather, the relative areas of red and blue are determined more by the height of the bars than the widths. On the other hand, the generalization of Benford’s law to second and later digits is not trivial, even for binary numbers. Using Benford’s Law to detect fraudulent scientific data.

### Détection de fraudes et loi de Benford : quelques risques associés – Munich Personal RePEc Archive

benord A method of accounting fraud detection based on bootstrapping and regression has been proposed. They lead instead to normal probability distributions again by the central limit theoremwhich do not satisfy Benford’s law. For instance, the probability that a “2” is encountered as the second digit is [36]. Thus, real-world distributions that span several orders of magnitude rather uniformly e.

The graph to the right shows Benford’s law for base Bbenford you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. The importance of this benchmark for detecting irregularities in prices was first demonstrated in a Europe-wide study [26] which investigated consumer price digits before and after the euro introduction for price adjustments.

An empirical distribution is called equivalent to benfoord Benford’s law if a distance for example total variation distance or the usual Euclidean distance between the probability mass bengord is sufficiently small. As a rule of thumb, the more orders of magnitude that the data evenly covers, bennford more accurately Benford’s law applies. If there is a list of lengths, the distribution of first digits of numbers in the list may be generally similar regardless of whether all the lengths are expressed in metres, or yards, or feet, or inches, etc.

He showed in a simulation study that long right-tailed distributions of a random variable are compatible with the Newcomb-Benford law, and that for distributions of the ratio of two random variables the fit generally improves.