General formulas for the central and non-central moments of the multinomial distribution

We present the first general formulas for the central and non-central moments of the multinomial distribution, using a combinatorial argument and the factorial moments previously obtained in Mosimann (1962). We use the formulas to give explicit expressions for all the non-central moments up to order 8 and all the central moments up to order 4. These results expand significantly on those in Newcomer (2008) and Newcomer et al. (2008), where the non-central moments were calculated up to order 4.


The multinomial distribution
For any d ∈ N, the d-dimensional (unit) simplex is defined by S := x ∈ [0, 1] d : d i=1 x i ≤ 1 , and the probability mass function k → P k,m (x) for ξ := (ξ 1 , ξ 2 , . . . , ξ d ) ∼ Multinomial(m, x) is defined by where m ∈ N and x ∈ S. If x d+1 := 1− d i=1 x i , then (1.1) is just a reparametrization of (ξ, 1− d i=1 ξ i ) ∼ Multinomial(m, (x, x d+1 )) where d+1 i=1 x i = 1. In this paper, our main goal is to give general formulas for the non-central and central moments of (1.1), namely p 1 , p 2 , . . . , p d ∈ N 0 . (1.2) We obtain the formulas using a combinatorial argument and the general expression for the factorial moments found in Mosimann (1962), which we register in the lemma below.
The formulas that we develop for the expectations in (1.2) will be used to compute explicitly all the non-central moments up to order 8 and all the central moments up to order 4, which expands on the third and fourth order non-central moments that were previously calculated in (Newcomer, 2008, Appendix A.1). We should also mention that explicit formulas for several lower-order (mixed) cumulants were presented in Wishart (1949) (see also (Johnson et al., 1997, p.37)), but not for the moments.

Results
First, we give a general formula of the non-central moments of the multinomial distribution in (1.1).
Theorem 1 (Non-central moments). Let ξ ∼ Multinomial(m, x). Then, for all p 1 , p 2 , . . . , p d ∈ N 0 , where p k denotes a Stirling number of the second kind (i.e., the number of ways to partition a set of p objects into k non-empty subsets).
Proof. We have the following well-known relation between the power p ∈ N 0 of a number x ∈ R and the falling factorials of x: See, e.g., (Graham et al., 1994, p.262). Apply this relation to every ξ pi i and use the linearity of the expectation to get 3) The conclusion follows from Lemma 1.
We can now deduce a general formula for the central moments of the multinomial distribution.

Numerical implementation
The formulas in Theorem 1 and Theorem 2 can be implemented in Mathematica as follows:

Explicit formulas
In Newcomer (2008), explicit expressions for the non-central moments of order 3 and 4 where obtained for the multinomial distribution, see also Newcomer et al. (2008); Ouimet (2020d). To expand on those results, we use the formula from Theorem 1 in the two subsections below to calculate (explicitly) all the non-central moments up to order 8 and all the central moments up to order 4.
Here is a table of the Stirling numbers of the second kind that we will use in our calculations:

Computation of the non-central moments up to order 8
By applying the general expression in Theorem 1 and by removing the Stirling numbers p i k i that are equal to 0, we get the following results directly.

Computation of the central moments up to order 4
With the results of the previous subsection and some algebraic manipulations (or the formula in Theorem 2), we can now calculate the central moments explicitly. We could calculate them up to order 8, but it would be very tedious. Instead, we write them up to order 4 for the sake of brevity. The simplifications we make to obtain the boxed expressions below are done with Mathematica.

Conclusion
In this short paper, we found general formulas for the central and non-central moments of the multinomial distribution as well as explicit formulas for all the non-central moments up to order 8 and all the central moments up to order 4. Our work expands on the results in Newcomer (2008), where the central moments were calculated up to order 4. It also complements the general formula for the (joint) factorial moments from Mosimann (1962) and the explicit formulas for some of the lower-order (mixed) cumulants that were presented in Wishart (1949).