BUILT-IN MATHEMATICA SYMBOL

Cumulant

gives the r

cumulant of the symbolic distribution dist.

Cumulant

gives the r

cumulant of the elements in the list.

Cumulant[r]
represents the r

formal cumulant.

MORE INFORMATION

For a symbolic distribution dist, the cumulant is given by the coefficient of in CumulantGeneratingFunction.

For a multivariate symbolic distribution dist, the cumulant is given by the series coefficient of in CumulantGeneratingFunction.

Cumulant 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 (5)

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:

In[1]:=

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Use symbolic data:

In[2]:=

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In[3]:=

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Compute the second cumulant of a continuous univariate distribution:

In[1]:=

Out[1]=

The cumulant

of a discrete univariate distribution:

In[1]:=

Out[1]=

The cumulant

for a multivariate distribution:

In[1]:=

Out[1]=

Find the relation of formal cumulant to formal moment:

In[1]:=

Out[1]=

Evaluate for a particular distribution:

In[2]:=

Out[2]=

In[3]:=

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:

TraditionalForm formatting:

Generalizations & Extensions (1)

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:

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 (1)

Find an unbiased estimator for a product of cumulants:

Check the sampling population expectation: