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Related Concept Videos

Weighted Mean00:57

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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.
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One-Way ANOVA can be performed on three or more samples with equal or unequal sample sizes. When one-way ANOVA is performed on two datasets with samples of equal sizes, it can be easily observed that the computed F statistic is highly sensitive to the sample mean.
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Given simple random samples of size n from a given population with a measured characteristic such as mean, proportion, or standard deviation for each sample, the probability distribution of all the measured characteristics is called a sampling distribution. How much the statistic varies from one sample to another is known as the sampling variability of a statistic. You typically measure the sampling variability of a statistic by its standard error. The standard error of the mean is an example...
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While measuring the mean of a data set, care needs to be taken when associating the mean to its central tendency. The same goes for the arithmetic mean, the geometric mean, or the harmonic mean. This is because the presence of a single outlier data value can significantly affect the mean. That is, the mean is sensitive to fluctuations in the data set.
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Stochastic Order and Generalized Weighted Mean Invariance.

Mateu Sbert1, Jordi Poch1, Shuning Chen2

  • 1Graphics and Imaging Laboratory, University of Girona, 17003 Girona, Spain.

Entropy (Basel, Switzerland)
|June 2, 2021
PubMed
Summary
This summary is machine-generated.

Order invariance is shown for weighted quasi-arithmetic means. When weight distributions are stochastically ordered, the means maintain the same order, simplifying analysis in diverse fields like economics and physics.

Keywords:
Kolmogorov meanRényi entropyShannon entropyTsallis entropyarithmetic meancross-entropyharmonic meanmultiple importance samplingquasi-arithmetic meanstochastic orderweighted mean

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Area of Science:

  • Mathematical Statistics
  • Information Theory
  • Economics

Background:

  • Weighted quasi-arithmetic means generalize classical means and are used across disciplines.
  • Stochastic orders on weights are crucial for risk analysis and advanced computational methods.

Purpose of the Study:

  • To establish theoretical results on order invariance for weighted quasi-arithmetic means.
  • To explore how stochastic ordering of weights impacts the ordering of these means.
  • To investigate invariance properties related to convex/concave functions and stochastic orders.

Main Methods:

  • Theoretical analysis of weighted quasi-arithmetic means and monotonic series.
  • Application of first stochastic order to weight distributions.
  • Exploration of invariance properties with convex/concave functions and a novel mirror property of stochastic orders.

Main Results:

  • Demonstrated that stochastically ordered weight distributions preserve the order of weighted quasi-arithmetic means for monotonic series.
  • Established alignment between arithmetic and harmonic means under stochastic ordering of weights.
  • Showcased applications in entropy, cross-entropy, and Monte Carlo methods.

Conclusions:

  • Order invariance theorems for weighted quasi-arithmetic means provide a framework for analyzing systems with changing weight distributions.
  • The findings are broadly applicable, enhancing understanding in fields utilizing generalized means and stochastic ordering.