This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# Cumulant

 Cumulant gives the r cumulant of the symbolic distribution dist. Cumulantgives the r cumulant of the elements in the list. Cumulant[r]represents the r formal cumulant.
• Cumulant is computed from the equivalent expression of sample moments.
• Cumulant handles both numerical and symbolic data.
Compute cumulants from data:
Use symbolic data:
Compute the second cumulant of a continuous univariate distribution:
The cumulant of a discrete univariate distribution:
The cumulant for a multivariate distribution:
Find the relation of formal cumulant to formal moment:
Evaluate for a particular distribution:
Compute cumulants from data:
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Use symbolic data:
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Compute the second cumulant of a continuous univariate distribution:
 Out[1]=

The cumulant of a discrete univariate distribution:
 Out[1]=

The cumulant for a multivariate distribution:
 Out[1]=

Find the relation of formal cumulant to formal moment:
 Out[1]=
Evaluate for a particular distribution:
 Out[2]=
 Out[3]=
 Scope   (8)
Compute cumulant of a univariate distribution:
Compute a cumulant of a specific order:
Evaluate a cumulant of a specific order numerically:
Compute cumulants of multivariate distributions:
Find cumulant of a user-defined distribution:
Compute cumulants of distributions derived from data:
Compute cumulant for a set of 5 independent identically distributed samples of size 1000:
Find joint cumulant from sparse data:
Compare with population moment:
 Applications   (3)
Estimate parameters of a distribution using the method of cumulants:
The law of large numbers states that a sample moment approaches the population moment as the sample size increases. Use Histogram to show the probability distribution of sample cumulant of standard normal random variates for different sample sizes:
Edgeworth's expansion of order :
Approximate SechDistribution:
First cumulant is equivalent to first moment :
Second cumulant is equivalent to the second central moment :
Third cumulant is equivalent to the third central moment :
Cumulant is equal to the derivative of the cumulant generating function at zero :
Use Cumulant directly:
Find the cumulant generating function using GeneratingFunction:
Sample estimator of Cumulant on data is biased:
Find a sampling population expectation, assuming size :
Construct an unbiased sample estimator using PowerSymmetricPolynomial:
Verify unbiasedness on a small sample size:
The sample estimator is biased:
Compare with the sampling population expectation of the sample estimator:
For some distributions with long tails, cumulants of only several low orders are defined:
Find an unbiased estimator for a product of cumulants:
Check the sampling population expectation:
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