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Related Concept Videos

Goodness-of-Fit Test01:16

Goodness-of-Fit Test

The goodness-of-fit test is a type of hypothesis test which determines whether the data "fits" a particular distribution. For example, one may suspect that some anonymous data may fit a binomial distribution. A chi-square test (meaning the distribution for the hypothesis test is chi-square) can be used to determine if there is a fit. The null and alternative hypotheses may be written in sentences or stated as equations or inequalities. The test statistic for a goodness-of-fit test is given as...
Expected Frequencies in Goodness-of-Fit Tests01:19

Expected Frequencies in Goodness-of-Fit Tests

A goodness-of-fit test is conducted to determine whether the observed frequency values are statistically similar to the frequencies expected for the dataset. Suppose the expected frequencies for a dataset are equal such as when predicting the frequency of any number appearing when casting a die. In that case, the expected frequency is the ratio of the total number of observations (n) to the number of categories (k).
Quantifying and Rejecting Outliers: The Grubbs Test01:02

Quantifying and Rejecting Outliers: The Grubbs Test

Sometimes, a data set can have a recorded numerical observation that greatly  deviates from the rest of the data. Assuming that the data is normally distributed, a statistical method called the Grubbs test can be used to determine whether the observation is truly an outlier.  To perform a two-tailed Grubbs test, first, calculate the absolute difference between the outlier and the mean. Then, calculate the ratio between this difference and the standard deviation of the sample. This number is...
Introduction to the Sign Test01:10

Introduction to the Sign Test

The sign test is an important tool in nonparametric statistics, offering a straightforward yet effective method for analyzing matched pairs, nominal data, or hypotheses concerning the median of a population. It transforms data points into positive or negative signs, avoiding the need for assumptions about data distribution and instead focusing on the direction of change. It is particularly valuable when data does not conform to the normal distribution requirements of many parametric tests. For...
Test for Homogeneity01:23

Test for Homogeneity

The goodness–of–fit test can be used to decide whether a population fits a given distribution, but it will not suffice to decide whether two populations follow the same unknown distribution. A different test, called the test for homogeneity, can be used to conclude whether two populations have the same distribution. To calculate the test statistic for a test for homogeneity, follow the same procedure as with the test of independence. The hypotheses for the test for homogeneity can be stated as...
Statistical Methods to Analyze Parametric Data: Student t-Test and Goodness-of-Fit Test01:09

Statistical Methods to Analyze Parametric Data: Student t-Test and Goodness-of-Fit Test

In parametric statistics, two fundamental tests stand out for their utility and wide application: the Student's t-test and goodness-of-fit tests. These tests provide researchers with a robust method for drawing insights from data, testing hypotheses, and making informed decisions based on their findings.
The Student's t-test is a statistical test that examines if there is a statistically significant difference between the means of two groups. This test is instrumental when dealing with data...

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A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments
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Pareto versus lognormal: a maximum entropy test.

Marco Bee1, Massimo Riccaboni, Stefano Schiavo

  • 1Department of Economics, University of Trento, Trento, Italy. marco.bee@unitn.it

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 21, 2011
PubMed
Summary
This summary is machine-generated.

Many real-world distributions exhibit lognormal behavior that transitions to a power-law (Pareto) tail. A new maximum entropy test can identify these power-law tails and underlying data-generating processes.

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Area of Science:

  • Statistical modeling and complex systems analysis.
  • Exploration of data distributions in natural and social sciences.

Background:

  • Many natural and social phenomena, including earthquake magnitudes, species abundance, and economic data, often follow lognormal distributions over a wide range.
  • However, these distributions frequently exhibit a power-law (Pareto) tail in their extreme percentiles, indicating a deviation from a pure lognormal model.
  • Identifying the precise data-generating process, especially when it deviates from standard models like lognormal or Pareto, is a significant challenge.

Purpose of the Study:

  • To introduce and validate a novel statistical test for detecting power-law tails in data distributions.
  • To utilize the principle of maximum entropy for identifying underlying data-generating processes.
  • To compare the efficacy of the maximum entropy approach against established methods for distribution analysis.

Main Methods:

  • Development of a hypothesis test for power-law tails based on the principle of maximum entropy.
  • Application of the maximum entropy method to analyze statistical distributions across various complex systems.
  • Comparative analysis of the maximum entropy approach with other widely used statistical testing methodologies.

Main Results:

  • The maximum entropy test successfully identifies the presence of power-law tails in distributions that appear lognormal over their main body.
  • This methodology can discern the true data-generating process, even when it is neither purely lognormal nor Pareto.
  • Results support the theory that distributions with a lognormal body and Pareto tail can arise from mixtures of lognormally distributed components.

Conclusions:

  • The maximum entropy framework provides a robust method for detecting power-law tails and characterizing complex data distributions.
  • This approach offers a powerful tool for distinguishing between different statistical models and understanding the underlying mechanisms of complex systems.
  • The findings contribute to the theoretical understanding of how mixed distributions with lognormal and Pareto characteristics emerge.