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Related Concept Videos

Fisher's Exact Test01:08

Fisher's Exact Test

857
Fisher's exact test is a statistical significance test widely used to analyze 2x2 contingency tables, particularly in situations where sample sizes are small. Unlike the chi-squared test, which approximates P-values and assumes minimum expected frequencies of at least five in each cell, Fisher's exact test calculates the exact probability (P-value) of observing the data or more extreme results under the null hypothesis. This feature makes it especially valuable when the assumptions of...
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Behrens–Fisher Test00:57

Behrens–Fisher Test

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The Behrens-Fisher test is a statistical method designed to address the Behrens-Fisher problem, which arises when comparing the means of two normally distributed populations with unequal variances. Unlike the Student's t-test, which assumes equal variances, the Behrens-Fisher test allows for mean comparison without this restrictive assumption. This flexibility makes it particularly valuable in scenarios where two independent samples exhibit normality but lack variance homogeneity.
This test...
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Poisson's Ratio01:23

Poisson's Ratio

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Poisson's ratio is a material property that indicates their stress response. It explains the connection between the elongation or compression a material undergoes in the direction of an applied force and the contraction or expansion it experiences perpendicular to that force. When a slender bar is loaded axially, it stretches in the direction of the force and contracts laterally. Poisson's ratio is the negative ratio of this lateral contraction to the axial elongation. The negative sign...
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F Distribution01:19

F Distribution

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The F distribution was named after Sir Ronald Fisher, an English statistician. The F statistic is a ratio (a fraction) with two sets of degrees of freedom; one for the numerator and one for the denominator. The F distribution is derived from the Student's t distribution. The values of the F distribution are squares of the corresponding values of the t distribution. One-Way ANOVA expands the t test for comparing more than two groups. The scope of that derivation is beyond the level of this...
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Relative Frequency Distribution00:55

Relative Frequency Distribution

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A relative frequency distribution is the proportion or fraction of times a value occurs in a data set. To find the relative frequencies, one can divide each frequency by the total number of data points in the sample. It is very similar to a regular frequency distribution, except that instead of reporting how many data values fall in a class, a relative frequency distribution reports the fraction of data values that fall in a class. These fractions or proportions are called relative frequencies...
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Weighted Mean00:57

Weighted Mean

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While taking the arithmetic, geometric, or harmonic mean of a sample data set, equal importance is assigned to all the data points. However, all the values may not always be equally important in some data sets. An intrinsic bias might make it more important to give more weightage to specific values over others.
For example, consider the number of goals scored in the matches of a tournament. While computing the average number of goals scored in the tournament, it may be more important to...
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Interval-valued optimization problems involving (α, ρ)-right upper-Dini-derivative functions.

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Weighted Relative Group Entropies and Associated Fisher Metrics.

Iulia-Elena Hirica1, Cristina-Liliana Pripoae2, Gabriel-Teodor Pripoae1

  • 1Faculty of Mathematics and Computer Science, University of Bucharest, Academiei 14, 010014 Bucharest, Romania.

Entropy (Basel, Switzerland)
|January 21, 2022
PubMed
Summary

New α-weighted group entropy functionals and Fisher-like metrics offer advanced tools for statistical models. A key finding links these metrics to canonical ones, revealing conditions for conformal relationships.

Keywords:
Fisher metricgroup Fisher metricrelative group entropyweighted entropyα-weighted group entropy

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Area of Science:

  • Information Geometry
  • Statistical Mechanics
  • Differential Geometry

Background:

  • Entropy functionals are crucial for quantifying information in statistical models.
  • Semi-Riemannian geometry provides a framework for analyzing statistical manifolds.
  • Existing metrics may not fully capture the complexities of weighted group entropies.

Purpose of the Study:

  • To introduce a novel class of α-weighted group entropy functionals.
  • To investigate associated Fisher-like metrics within a geometric framework.
  • To establish a relationship between these new metrics and canonical metrics.

Main Methods:

  • Definition of α-weighted group entropy functionals.
  • Construction of associated Fisher-like metrics.
  • Application of semi-Riemannian geometry principles.
  • Analysis of metric conformality and (homothetic) properties.

Main Results:

  • A new family of α-weighted group entropy functionals and their metrics are presented.
  • A significant link is established between the introduced metric and a canonical metric.
  • A sufficient, universal condition for metric conformality (or homothety) is derived.
  • A known result for α=1 and non-weighted entropies is recovered.

Conclusions:

  • The developed α-weighted group entropy functionals and metrics are effective geometric tools for statistical models.
  • The established conformality condition provides a deeper understanding of the metric's behavior.
  • These findings offer a generalized approach applicable to various entropy-related statistical structures.