MomCumConvert: A Package for Conversion Between Moments and Cumulants

Qi Zheng
National Center for Toxicological Research
Jefferson, Arkansas 72079
April 20, 2001

MomCumConvert is a package designed  to produce formulae expressing statistical moments in terms of cumulants and vice versa. Two methods have been implemented in the package, a method based on Kendall's  operator and an elementary method of equating coefficients of generating functions. The two methods produce the same formulae, but speed can vary depending on the particular formula to be generated. For implementation details, the reader is referred to Zheng (1998).

<< MomCumConvert.m

There is only one user function in this package: MomCumConvert[]. But five options of the function enable the user to choose what types of formulae to generate and which method to be used to generate them.

?MomCumConvert

Options[MomCumConvert]

To demonstrate the use of MomCumConvert, we first produce the well-known formula Var[
              X] = E[(X^2)] - (((E[X]))^2).

MomCumConvert[{2}, ForMoment -> False, Centered -> False]

We  can use 2 formatting rules to display the formulae in standard notation.

Clear[K, M];
Format[M[i___]] := Subscript[μ, i];
Format[K[i___]] := Subscript[κ, i];

MomCumConvert[{2}, ForMoment -> False, Centered -> False, 
      MomentSymbol -> M, CumulantSymbol -> K]

Now we produce the last formula in equation (3.81) from the first volume of Kendall's Advanced Theory of Statistics (Stuart and Ord,1994).

MomCumConvert[{3, 3}, ForMoment -> False, MomentSymbol -> M, 
      CumulantSymbol -> K]

As the following example demonstrates, the two methods always produce  the same formula.

MomCumConvert[{3, 1, 3}, ForMoment -> False, MomentSymbol -> M, 
      CumulantSymbol -> K, KendallMethod -> True]

MomCumConvert[{3, 1, 3}, ForMoment -> False, MomentSymbol -> M, 
      CumulantSymbol -> K, KendallMethod -> False]

Finally, we generate a formula for crude (noncentral) moment, which appeared on Page 3 of Kratky et al. (1972).

MomCumConvert[{3, 1, 1}, Centered -> False, MomentSymbol -> M, 
      CumulantSymbol -> K]

References

A. Stuart and J.K. Ord, Kendall's Advanced Theory of Statistics, Vol. 1: Distribution Theory, Sixth edition, Edward Arnold, London, 1994.

J. Kratky, J. Reinfelds, K. Hutcheson and L.R. Shenton, Tables of crude moments expressed in  terms of cumulants, Technical report, The University of Georgia, Athens, Georgia, 1972.

Q. Zheng, Computing relations between statistical moments and cumulants,  Proceedings of the Computational Statistics Section, American Statistical Association, pp. 165-170, 1998.