Multivariate probability distribution

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SUMMARY

In multivariate probability distribution, higher-order cumulants provide diminishing significance compared to higher-order moments. For normal distributions, cumulants above the second order are zero, indicating they do not contribute any information. The first two cumulants represent the means and covariances, which are critical for understanding the distribution's characteristics. It is essential to recognize that while higher-order moments contain information, they cannot be set to zero without violating mathematical properties.

PREREQUISITES
  • Understanding of multivariate probability distributions
  • Knowledge of cumulants and moments in statistics
  • Familiarity with normal distribution properties
  • Basic concepts of covariance and correlation
NEXT STEPS
  • Study the properties of cumulants in multivariate distributions
  • Learn about the implications of higher-order moments in statistical analysis
  • Explore the mathematical proofs related to cumulants and moments
  • Investigate applications of multivariate probability distributions in real-world scenarios
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Statisticians, data scientists, and researchers involved in advanced statistical modeling and analysis of multivariate data distributions.

beman
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"In multivariate probability distribution higher-order cumulants contain information of decreasing significance, unlike higher-order moments".
 
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Can you provide more detail? is this for a course? did you read it in some book? what is the context of your question?
 
For normal distributions cumulants above those of second order vanish, which pretty much means they don't contribute much information.
 
Last edited:
In multivariate probability distribution the first two cumulants are the means and covariances.Higher-order cumulants contain information of decreasing significance, unlike higher-order moments.We cannot set all moments higher than a certain order equal to zero since E(X^2n)>=E(X^n)^2 and thus,all moments contain information about
the lower moments.
 
beman said:
In multivariate probability distribution the first two cumulants are the means and covariances.Higher-order cumulants contain information of decreasing significance, unlike higher-order moments.We cannot set all moments higher than a certain order equal to zero since E(X^2n)>=E(X^n)^2 and thus,all moments contain information about
the lower moments.

Again - for normal distributions, cumulants of order higher than two are zero: it isn't decreasing significance, it is no significance.
 

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