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

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Basic Examples(2)

Compute cumulants from data:

Use symbolic data:

Compute the second cumulant of a continuous univariate distribution:

The cumulant for a multivariate distribution:

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:

Work with large arrays:

SparseArray data can be used just like dense arrays:

Find cumulants of WeightedData:

Find a cumulant of EventData:

Find a cumulant of TimeSeries:

The cumulant depends only on the values:

Find a cumulant for data involving quantities:

Distribution and Process Cumulants(5)

Find the cumulants for univariate distributions:

Multivariate 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:

Data distribution:

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)

Convert combinations of formal moments to an expression involving Cumulant:

Evaluate an expression involving formal cumulants for a distribution:

Evaluate for data:

Find a sample estimator for an expression involving Cumulant:

Evaluate the resulting estimator for data:

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:

Use the window of length .1:

Compute cumulants for slices of a collection of paths of a random process:

Choose a few slice times:

Plot cumulants over these paths:

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)

For some distributions with long tails, cumulants of only several low orders are defined:

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: