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)

TraditionalForm formatting for formal cumulants:

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: