Cumulant
Cumulant[dist,r]
gives the r cumulant of the distribution dist.
Cumulant[list,r]
gives the r cumulant of the elements in the list.
Cumulant[r]
represents the r formal cumulant.
Details

- For a distribution dist, the r
cumulant is given by the coefficient of
in CumulantGeneratingFunction[dist,t].
- For a multivariate distribution dist, the {r1,r2,…}
cumulant is given by the series coefficient of
in CumulantGeneratingFunction[dist,{t1,t2,…}].
- Cumulant[list,r] is computed from the equivalent expression of sample moments.
- Cumulant handles both numerical and symbolic data.
- Cumulant works with SparseArray objects.
- Cumulant[r] can be used in such functions as MomentConvert and MomentEvaluate, etc.
Examples
open allclose allBasic Examples (2)
Scope (18)
Data Cumulants (9)
Exact input yields exact output:
Approximate input yields approximate output:
Cumulant for a matrix gives column-wise means:
Cumulant for a tensor gives column-wise means at the first level:
SparseArray data can be used just like dense arrays:
Find cumulants of WeightedData:
Find a cumulant of EventData:
Find a cumulant of TimeSeries:
Distribution and Process Cumulants (5)
Find the cumulants for univariate distributions:
Compute a cumulant for a symbolic order r:
A cumulant may only evaluate for specific orders:
A cumulant may only evaluate numerically:
Cumulants for derived distributions:
Cumulant function for a random process:
Find a cumulant of TemporalData at some time t=0.5:
Find the corresponding cumulant function together with all the simulations:
Formal Cumulants (4)
TraditionalForm formatting for formal cumulants:
Convert combinations of formal moments to an expression involving Cumulant:
Evaluate an expression involving formal cumulants for a distribution:
Find a sample estimator for an expression involving Cumulant:
Applications (5)
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:
Compute a moving cumulant for some data:
Compute cumulants for slices of a collection of paths of a random process:
Properties & Relations (4)
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:
Check using CumulantGeneratingFunction:
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:
Possible Issues (1)
Neat Examples (2)
Find an unbiased estimator for a product of cumulants:
Check the sampling population expectation:
The distribution of Cumulant estimates for 20, 100, and 300 samples:
Text
Wolfram Research (2010), Cumulant, Wolfram Language function, https://reference.wolfram.com/language/ref/Cumulant.html.
CMS
Wolfram Language. 2010. "Cumulant." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Cumulant.html.
APA
Wolfram Language. (2010). Cumulant. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Cumulant.html