Unbiased Estimation of Population Moments;
Moments of Moments

mathStatica can find unbiased estimators of population moments. For instance, it offers h-statistics (unbiased estimators of population central moments), k-statistics (unbiased estimators of population cumulants), multivariate varieties of the same, polykays (unbiased estimators of products of cumulants) and more. Consider the k-statistic which is an unbiased estimator of the cumulant ; that is, , for . Here are the are k-statistics:

As per convention, the solution is expressed in terms of power sums .

Moments of moments: Because the above expressions (sample moments) are functions of random variables , we might want to calculate population moments of them. With mathStatica, we can find any moment (raw, central, or cumulant) of the above expressions. For instance, is meant to have the property that . We test this by calculating the first raw moment of , and express the answer in terms of cumulants:

In 1928, Fisher published the product cumulants of the k-statistics, which are now listed in reference bibles such as Stuart and Ord (1994). Here is the solution to :

This is the correct solution. Unfortunately, the solutions given in Stuart and Ord (1994, equation (12.70)) and Fisher (1928) are actually incorrect.