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: