BUILT-IN MATHEMATICA SYMBOL

# MomentConvert

MomentConvert[mexpr, form]
converts the moment expression mexpr to the specified form.

## DetailsDetails

• MomentConvert handles both formal moment and formal sample moment expressions.
• A formal moment expression can be any polynomial in formal moments 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
• Formal moment expressions can be evaluated for any particular distribution using MomentEvaluate.
• A moment expression can be converted into any other moment expression.
• The following forms can used for converting between moment expressions:
•  "Moment" convert to formal moments "CentralMoment" convert to formal central moments "FactorialMoment" convert to formal factorial moments "Cumulant" convert to formal cumulants
• A sample moment expression is any polynomial in formal symmetric polynomials of the form:
•  PowerSymmetricPolynomial[r] formal r power symmetric polynomial AugmentedSymmetricPolynomial[{r1,r2,...}] formal augmented symmetric polynomial
• Sample moment expressions can be evaluated on a dataset using MomentEvaluate.
• A sample moment expression can be converted into any other sample moment expression.
• The following forms can used for converting between sample moment expressions:
•  "PowerSymmetricPolynomial" convert to formal power symmetric polynomial "AugmentedSymmetricPolynomial" convert to formal augmented symmetric polynomial
• Sample moment expressions are effectively moment estimators assuming independent, identically distributed samples.
• Moment estimators for a given moment expression can be constructed using the forms:
•  "SampleEstimator" construct a sample moment estimator "UnbiasedSampleEstimator" construct an unbiased sample moment estimator
• Sample moment expressions can be considered a random variable constructed from independent, identically distributed random variables. The expected value can be found by converting from its sample moment expression to a moment expression.
• The expectation for a given sample moment expression can be computed using the forms:
•  "Moment" express in terms of formal moments "CentralMoment" express in terms of formal central moments "FactorialMoment" express in terms of formal factorial moments "Cumulant" express in terms of formal cumulants
• MomentConvert[expr, form1, form2, ...] first converts to , then to , etc.

## ExamplesExamplesopen allclose all

### Basic Examples (2)Basic Examples (2)

Express the cumulant in terms of raw moments:

Express the multivariate cumulant in terms of central moments:

Find an unbiased sample estimator for the second cumulant, i.e. second k-statistics:

Convert the estimator to the basis of power symmetric polynomials:

Compute expectation of the estimator in terms of raw moments:

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