# MomentEvaluate

MomentEvaluate[mexpr,dist]

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

MomentEvaluate[mexpr,list]

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

MomentEvaluate[mexpr,dist,list]

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

# Details • 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 {r1,r2,…} augmented symmetric polynomial
• For a sample moment expression is taken to be the length of the list of data.
• MomentEvaluate[mexpr,,n] 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:

## 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(2)

Find expectation of estimator on a sample from Bernoulli distribution:

Express the expectation of the estimator in terms of formal moments:

Expectation of the estimator for a sample from Bernoulli distribution:

Variance of the sample estimator:

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:

Introduced in 2010
(8.0)