BUILT-IN MATHEMATICA SYMBOL

MomentEvaluate

evaluates formal moments in the moment expression mexpr on the distribution dist.

MomentEvaluate

evaluates formal moments and formal sample moments in mexpr on the data list.

MomentEvaluate

evaluates formal moments on the distribution dist and formal sample moments on the data list.

MORE INFORMATION

A moment expression is an expression involving formal moments and formal sample moments.

A formal moment is an expression of the form:

	Moment[r]	formal r moment
	CentralMoment[r]	formal r central moment
	FactorialMoment[r]	formal r factorial moment
	Cumulant[r]	formal r cumulant

A formal sample moment is an expression of the form:

	PowerSymmetricPolynomial[r]	formal r power symmetric polynomial
	AugmentedSymmetricPolynomial[{r₁,r₂,...}]	formal augmented symmetric polynomial

For a sample moment expression PowerSymmetricPolynomial is taken to be the length of the list of data.

MomentEvaluate assumes that n is taken to be the length of the list of data.

EXAMPLES

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Basic Examples (3)

Evaluate formal moments for a univariate distribution:

Evaluate formal moments for a multivariate distribution:

Evaluate sample formal moments for data:

Evaluate formal moments for data:

Evaluate formal moments for a univariate distribution:

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Evaluate formal moments for a multivariate distribution:

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Evaluate sample formal moments for data:

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Evaluate formal moments for data:

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Scope (6)

Evaluate mixed univariate formal moment polynomial for a distribution:

Evaluate mixed multivariate formal moment polynomial for a distribution:

Evaluate polynomial in formal moments for data:

Compare with direct evaluation:

Evaluate formal sample polynomial for data:

Evaluate formal sample polynomial for data with n being the sample size:

Evaluate an expression containing both formal moments and formal sample moments:

Alternatively:

Generalizations & Extensions (1)

Compute mean, variance, skewness, and excess kurtosis expressed in terms of Cumulant:

Compare with direct evaluation:

Applications (1)

Construct sample and unbiased estimators for

:

Accumulate statistics of these estimators on the same data:

Compare the means of these statistics with population cumulant:

Find sampling population expectation of estimators for distribution dist:

Find sampling population variance of estimators for distribution dist:

Numerically evaluate expected variances for sample sizes used:

Compare to sample values:

Properties & Relations (1)

MomentEvaluate effectively evaluates a moment expression by evaluating its constituents: