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MomentConvert

converts the moment expression mexpr to the specified form.

MORE INFORMATION

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 first converts to , then to , etc.

EXAMPLES

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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:

Express the cumulant in terms of raw moments:

In[1]:=

Out[1]//TraditionalForm=

Express the multivariate cumulant in terms of central moments:

In[2]:=

Out[2]//TraditionalForm=

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

In[1]:=

Out[1]//TraditionalForm=

Convert the estimator to the basis of power symmetric polynomials:

In[2]:=

Out[2]//TraditionalForm=

Compute expectation of the estimator in terms of raw moments:

In[3]:=

Out[3]=

Scope (4)

Express a multivariate cumulant in terms of raw moments:

Convert back to the cumulant:

Find an unbiased sample estimator for a product of univariate central moments, also known as polyache:

Find a multivariate polyache:

Find the sampling distribution estimator expectation of an augmented symmetric polynomial:

Convert the augmented symmetric polynomial to the basis of power symmetric polynomials:

Evaluate at a sample of size 3:

Compare to direct evaluation of the augmented symmetric polynomial:

Generalizations & Extensions (2)

Find the h-statistic in basis of power symmetric polynomials:

Find the sampling distribution expectation of a sample estimator of the third central moment:

Applications (21)

Convert raw moments to central moments:

Factorial moments:

Cumulants:

Show all cross-conversions between formal moments:

Show multivariate cross-conversions between formal moments:

Compute a sample estimator of the variance:

Compute the bias of the estimator as the mean with respect to the sampling population distribution, assuming a sample of size

:

Illustrate the computation by evaluating the estimator on a symbolic sample of size 5:

Now compute its expectation assuming

are independent random variates from normal distribution:

Compare with the result obtained above:

Compute the expected variance of a sample estimator of the variance:

Compute the variance as the second central moment of the estimator:

In the large sample size limit, the variance of the estimator tends to zero in agreement with the law of large numbers:

Perform 1000 simulations using standard normal samples of size 30:

Compare the sample mean and variance to their expected values:

Compute covariance between the sample mean and sample variance estimators:

Compute sampling population covariance as a mixed central moment:

The expected covariance vanishes on normal samples:

Find a sample estimator of the off-diagonal covariance matrix element of two-dimensional data:

Find its bias and its variance:

Compute the bias and the variance of the estimator for binormal samples:

Estimate the sample size needed for the variance of the estimator on standard binormal samples with

not to exceed 0.001:

The sample estimator of the standard deviation is computed as the square root of the sample variance:

Such an estimator is biased and underestimates the population standard deviation:

The analysis of the standard deviation estimator is carried out by replacing the nonlinear function with its truncated Taylor series about the bias of its argument:

Find the expectation of the approximated estimator:

Compute its numerical value for a standard normal sample of size

:

Find the variance of the estimator for normal samples:

Derive finite-sample Jarque-Bera statistics

:

Find the mean and variance of sample skewness estimator

on size

normal samples:

Compute the mean and variance of sample kurtosis estimator

on size

normal samples:

Assemble the estimator:

Find large

approximation:

Sample moment estimators are automatically unbiased:

Compute an unbiased moment estimator in terms of power symmetric polynomials:

Compute the sampling population expectation of the estimator:

Compute the multivariate moment estimator:

These are also unbiased:

Evaluate the estimator on a symbolic sample:

Factorial moments can be expressed as the linear combination of raw moments:

Hence their sample estimators are automatically unbiased as well:

Compute an unbiased factorial moment estimator in terms of power symmetric polynomials:

Compute the sampling population expectation of the estimator:

Find the second h-statistics:

Write the h-statistics in terms of power symmetric polynomials:

Compare it with the sample estimator of the second central moment

:

Find the sampling population expectation of these estimators for sample size

:

Compute the third h-statistics in terms of power symmetric polynomials:

Compare it with the sample estimator of central moment

:

Find the sampling population expectation of these estimators for sample size

:

Find multivariate h-statistics for

:

Evaluate the estimator on a sample from a binormal distribution:

Compare with the population value:

Find fourth k-statistics in terms of power symmetric polynomials:

Evaluate obtained k-statistics on a standard normal sample:

Accumulate statistics of the estimator and show the histogram:

Compute multivariate k-statistics for

:

Compare it to the sample estimator:

Find the unbiased estimator of the second power of mean:

Evaluate it on a symbolic sample:

Find the sample population expectation:

Compute the unbiased estimator of the product of cumulants, also known as polykay:

Express it in terms of power symmetric polynomials:

Find the unbiased estimator of the product of multivariate central moments, also known as polyache:

Find the value of the estimator on a multivariate sample:

Compare with sampling population moments:

Cumulants of k-statistics are polynomials in sampling population expectations of certain monomials of k-statistics. They are built using umbral calculus, starting with expression of the multivariate cumulant in terms of raw moments:

Each multivariate moment is understood as the sampling population expectation of the monomial in k-statistics. For instance, raw moment

stands for the product of expectation of

. Find the resulting unbiased estimator for

and

:

Define a procedure for computation of cumulants of k-statistics:

Verify that

:

Verify that

:

This implies that the sample mean and sample variance of a normal sample are independent:

Cumulants of k-statistics were tabulated because they were thought to give more concise expressions, and were used for computation of moments of estimators. Compute the cumulant of second k-statistics:

Compute the cumulant of the product of k-statistics:

Expressions for higher-order cumulants of k-statistics quickly become big:

Compute the number of partitions of a set into subsets of given sizes:

There are 10 ways to partition the set of 5 elements into subsets of 2 and 3 elements:

Construct partitions and count directly:

partitions a list into blocks of specified lengths:

Properties & Relations (5)

The binomial theorem defines relations between formal moments and formal central moments:

Express formal factorial moments in terms of formal moments using Stirling numbers:

Polynomial in moments rewritten in terms of central moments may still involve the mean

:

The sample estimator of factorial moment is unbiased:

Compute cumulants through series expansion of logarithm of moment-generating function:

Possible Issues (2)

Conversion between forms of symmetric polynomials treats formal moments as constants:

Expressions involving AugmentedSymmetricPolynomial and PowerSymmetricPolynomial are converted:

MomentConvert requires input to be polynomial in formal and/or sample moments:

Neat Examples (2)

Cross convert between any pairs of formal moments:

Cross convert between any pairs of multivariate formal moments: