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^(th) moment
    CentralMoment[r]formal r^(th) central moment
    FactorialMoment[r]formal r^(th) factorial moment
    Cumulant[r]formal r^(th) 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^(th) 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:

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Express the multivariate cumulant in terms of central moments:

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

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Convert the estimator to the basis of power symmetric polynomials:

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Compute expectation of the estimator in terms of raw moments:

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