Unbiased estimators 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 much more.

Example:     k-statistics

Consider the k-statistic MomentsOfMoments_1.gif which is an unbiased estimator of the rth cumulant MomentsOfMoments_3.gif; that is, MomentsOfMoments_4.gif, for r = 1, 2, … .
Here are the 2nd are 3rd k-statistics:

In[1]:= MomentsOfMoments_5.gif

Out[1]= MomentsOfMoments_6.gif

Out[2]= MomentsOfMoments_7.gif

As per convention, the solution is expressed in terms of power sums MomentsOfMoments_8.gif.

Example:     Moments of moments

Because the above expressions (sample moments) are functions of random variables MomentsOfMoments_9.gif, we might want to calculate population moments of them. With mathStatica, we can find any moment (raw, central, or cumulant) of completley arbitrary expressions. For instance, MomentsOfMoments_10.gif is meant to have the property that MomentsOfMoments_11.gif. We test this by calculating the first raw moment of MomentsOfMoments_12.gif, and express the answer in terms of cumulants:

In[3]:= MomentsOfMoments_13.gif

Out[3]= MomentsOfMoments_14.gif

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 MomentsOfMoments_15.gif:

In[4]:= MomentsOfMoments_16.gif

Out[4]= MomentsOfMoments_17.gif

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

Example:    Variance of the 3rd sample central moment

Problem:  Find the variance of the 3rd sample central moment MomentsOfMoments_19.gif.
Solution:  We proceed in two steps:

Step [1]: Express the 3rd sample central moment in terms of power sums MomentsOfMoments_8.gif:

In[1]:= MomentsOfMoments_21.gif

Out[1]= MomentsOfMoments_22.gif

Step[2]: Since MomentsOfMoments_23.gif, the desired solution is simply the 2nd Central Moment of MomentsOfMoments_25.gif.

In[2]:= MomentsOfMoments_26.gif

Out[17]= MomentsOfMoments_27.gif

Example:  Multivariate       New in mathStatica 2.0

Problem:      Find  the covariance between MomentsOfMoments_28.gif  and   MomentsOfMoments_29.gif .
        Give the solution in terms of bivariate central moments of the population.

We proceed in two steps:

Step [1]: Enter each expression in terms of bivariate power sums  MomentsOfMoments_30.gif.
      The first expression MomentsOfMoments_31.gif.   The second expression MomentsOfMoments_32.gif.

Step[2]: Since Cov(a, b) =  MomentsOfMoments_33.gif, the desired solution is simply:

In[1]:= MomentsOfMoments_34.gif

Out[1]= MomentsOfMoments_35.gif

Example:    Bring out the BFG           New in mathStatica 2.0

Cook (1951, pp.187–195) derived explicit results for product cumulants of various bivariate k-statistics; some of the simpler results are listed in Stuart and Ord (1994, Section 13.3). In this example, we illustrate not only how to obtain these known and published results, but more generally how one can obtain any such desired product cumulant … not just the simpler cases that are already known / published.

To illustrate, we will work here with the bivariate k-statistics MomentsOfMoments_36.gif and MomentsOfMoments_37.gif:

In[1]:= MomentsOfMoments_38.gif

Out[1]= MomentsOfMoments_39.gif


In[2]:= MomentsOfMoments_40.gif

Out[2]= MomentsOfMoments_41.gif

We will now find the product cumulant MomentsOfMoments_42.gif. In  the notation of Cook (1951, p.190) and Stuart and Ord (1994, equation (13.9)), this is the expression  MomentsOfMoments_43.gif:

In[3]:= MomentsOfMoments_44.gif

Out[3]= MomentsOfMoments_45.gif

Here is MomentsOfMoments_46.gif;  in  the notation of Cook (1951, p.191), this is the expression MomentsOfMoments_47.gif:

In[4]:= MomentsOfMoments_48.gif

Out[4]= MomentsOfMoments_49.gif