General Formulas for the Central and Non-Central Moments of the Multinomial Distribution

Abstract

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.

1. The Multinomial Distribution

For any $d\in \mathbb{N}$, the d-dimensional (unit) simplex is defined by $\mathcal{S}:=\left\{\mathit{x}\in {[0,1]}^d:{\sum }_{i=1}^d{x}_i\le 1\right\}$, and the probability mass function $\mathit{k}\mapsto P_{\mathit{k},m}(\mathit{x})$ for $\xi :=({\xi }_1,{\xi }_2,\dots ,{\xi }_d)\sim \mathrm{Multinomial}(m,\mathit{x})$ is defined by

$$P_{\mathit{k},m}(\mathit{x}):=\frac{m!}{(m-{\sum }_{i=1}^d{k}_i)!{\prod }_{i=1}^d{k}_i!}·{(1-\sum _{i=1}^d{x}_i)}^{m-{\sum }_{i=1}^d{k}_i}\prod _{i=1}^d{x}_i^{k_i},\mathit{k}\in {\mathbb{N}}_0^d\cap m\mathcal{S},$$

(1)

where $m\in \mathbb{N}$ and $\mathit{x}\in \mathcal{S}$. If $x_{d+1}:=1-{\sum }_{i=1}^d{x}_i$, then (1) is just a reparametrization of $(\xi ,1-{\sum }_{i=1}^d{\xi }_i)\sim \mathrm{Multinomial}(m,(\mathit{x},x_{d+1}))$ where ${\sum }_{i=1}^{d+1}{x}_i=1$. In this paper, our main goal is to give general formulas for the non-central and central moments of (1), namely

$$\mathbb{E}\left[\prod _{i=1}^d{\xi }_i^{p_i}\right]\mathrm{and}\mathbb{E}\left[\prod _{i=1}^d{({\xi }_i-\mathbb{E}[{\xi }_i])}^{p_i}\right],p_1,p_2,\dots ,p_d\in {\mathbb{N}}_0.$$

(2)

We obtain the formulas using a combinatorial argument and the general expression for the factorial moments found in Mosimann (1962) [1], which we register in the lemma below.

Lemma 1

(Factorial moments). Let $\xi \sim \mathrm{Multinomial}(m,\mathit{x})$. Then, for all $k_1,k_2,\dots ,k_d\in {\mathbb{N}}_0$,

$$\mathbb{E}\left[\prod _{i=1}^d{\xi }_i^{(k_i)}\right]=m^{({\sum }_{i=1}^d{k}_i)}\prod _{i=1}^d{x}^{k_i},$$

(3)

where $m^{(k)}:=m(m-1)\dots (m-k+1)$ denotes the k-th order falling factorial of m.

The formulas that we develop for the expectations in (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) [2], Appendix A.1). We should also mention that explicit formulas for several lower-order (mixed) cumulants were presented in Wishart (1949) [3] (see also Johnson et al. (1997) [4], page 37), but not for the moments.

2. Motivation

To the best of our knowledge, general formulas for the central and non-central moments of the multinomial distribution have never been derived in the literature. The central moments can arise naturally, for example, when studying asymptotic properties, via Taylor expansions of statistical estimators involving the multinomial distribution. For a given sequence of i.i.d. observations ${\mathit{X}}_1,{\mathit{X}}_2,\dots ,{\mathit{X}}_n$, two examples of such estimators are the Bernstein estimator for the cumulative distribution function

$$F_{n,m}^{\star }(\mathit{x}):=\sum _{\mathit{k}\in {\mathbb{N}}_0^d\cap m\mathcal{S}}\frac{1}{n}\sum _{i=1}^n{𝟙}_{(-\infty ,\frac{\mathit{k}}{m}]}({\mathit{X}}_i)P_{\mathit{k},m}(\mathit{x}),\mathit{x}\in \mathcal{S},m,n\in \mathbb{N},$$

(4)

and the Bernstein estimator for the density function (also called smoothed histogram)

$${\widehat{f}}_{n,m}(\mathit{x}):=\sum _{\mathit{k}\in {\mathbb{N}}_0^d\cap (m-1)\mathcal{S}}\frac{m^d}{n}\sum _{i=1}^n{𝟙}_{(\frac{\mathit{k}}{m},\frac{\mathit{k}+1}{m}]}({\mathit{X}}_i)P_{\mathit{k},m-1}(\mathit{x}),\mathit{x}\in \mathcal{S},m,n\in \mathbb{N},$$

(5)

over the d-dimensional simplex. Some of their asymptotic properties were investigated by Vitale (1975), Stadtmüller (1986), Tenbusch (1997), Petrone (1999), Ghosal (2001), Petrone and Wasserman (2002), Babu et al. (2002), Kakizawa (2004), Bouezmarni and Rolin (2007), Bouezmarni et al. (2007), Leblanc (2009, 2010, 2012), Curtis and Ghosh (2011), Igarashi and Kakizawa (2014), Turnbull and Ghosh (2014), Lu (2015), Guan (2016, 2017) and Belalia et al. (2017, 2019) [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30] when $d=1$, by Tenbusch (1994) [31] when $d=2$, and by Ouimet (2020) [32,33] for all $d\ge 1$, using a local limit theorem from Ouimet (2020) [34] for the multinomial distribution (see also Arenbaev (1976) [35]). The estimator (5) is a discrete analogue of the Dirichlet kernel estimator introduced by Aitchison and Lauder (1985) [36] and studied theoretically in Brown and Chen (1999), Chen (1999, 2000) and Bouezmarni and Rolin (2003) [37,38,39,40] when $d=1$ (among others), and in Ouimet (2020) [41] for all $d\ge 1$.

3. Results

First, we give a general formula of the non-central moments of the multinomial distribution in (1).

Theorem 1

(Non-central moments). Let $\xi \sim \mathrm{Multinomial}(m,\mathit{x})$. For all $p_1,p_2,\dots ,p_d\in {\mathbb{N}}_0$,

$$\mathbb{E}\left[\prod _{i=1}^d{\xi }_i^{p_i}\right]=\sum _{k_1=0}^{p_1}\cdots \sum _{k_d=0}^{p_d}{m}^{({\sum }_{i=1}^d{k}_i)}\prod _{i=1}^d\left\{\genfrac{}{}{0.0pt}{}{p_i}{k_i}\right\}{x}_i^{k_i},$$

(6)

where $\left\{\genfrac{}{}{0.0pt}{}{p}{k}\right\}$ 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 $p\in {\mathbb{N}}_0$ of a number $x\in \mathbb{R}$ and the falling factorials of x:

$$x^p=\sum _{k=0}^p\left\{\genfrac{}{}{0.0pt}{}{p}{k}\right\}{x}^{(k)}.$$

(7)

See, e.g., (Graham et al. (1994) [42], page 262). Apply this relation to every ${\xi }_i^{p_i}$ and use the linearity of the expectation to get

$$\mathbb{E}\left[\prod _{i=1}^d{\xi }_i^{p_i}\right]=\sum _{k_1=0}^{p_1}\cdots \sum _{k_d=0}^{p_d}\left\{\genfrac{}{}{0.0pt}{}{p_1}{k_1}\right\}\dots \left\{\genfrac{}{}{0.0pt}{}{p_d}{k_d}\right\}\mathbb{E}\left[\prod _{i=1}^d{\xi }_i^{(k_i)}\right].$$

(8)

The conclusion follows from Lemma 1.    □

We deduce a general formula for the central moments of the multinomial distribution.

Theorem 2

(Central moments). Let $\xi \sim \mathrm{Multinomial}(m,\mathit{x})$. For all $p_1,p_2,\dots ,p_d\in {\mathbb{N}}_0$,

$$\mathbb{E}\left[\prod _{i=1}^d{({\xi }_i-\mathbb{E}[{\xi }_i])}^{p_i}\right]=\sum _{{\ell }_1=0}^{p_1}\cdots \sum _{{\ell }_d=0}^{p_d}\sum _{k_1=0}^{{\ell }_1}\cdots \sum _{k_d=0}^{{\ell }_d}{m}^{({\sum }_{i=1}^d{k}_i)}{(-m)}^{{\sum }_{i=1}^d(p_i-{\ell }_i)}\prod _{i=1}^d\left(\genfrac{}{}{0pt}{}{p_i}{{\ell }_i}\right)\left\{\genfrac{}{}{0.0pt}{}{{\ell }_i}{k_i}\right\}{x}_i^{p_i-{\ell }_i+k_i},$$

(9)

where $\left(\genfrac{}{}{0pt}{}{p}{\ell }\right)$ denotes the binomial coefficient $\frac{p!}{\ell !(p-\ell )!}$.

Proof. 

By applying the binomial formula to each factor ${({\xi }_i-\mathbb{E}[{\xi }_i])}^{p_i}$ and using the fact that $\mathbb{E}[{\xi }_i]=m{x}_i$ for all $i\in \{1,2,\dots ,d\}$, note that

$$\mathbb{E}\left[\prod _{i=1}^d{({\xi }_i-\mathbb{E}[{\xi }_i])}^{p_i}\right]=\sum _{{\ell }_1=0}^{p_1}\cdots \sum _{{\ell }_d=0}^{p_d}\mathbb{E}\left[\prod _{i=1}^d{\xi }_i^{{\ell }_i}\right]·\prod _{i=1}^d\left(\genfrac{}{}{0pt}{}{p_i}{{\ell }_i}\right){(-m{x}_i)}^{p_i-{\ell }_i}.$$

(10)

The conclusion follows from Theorem 1.    □

4. Numerical Implementation

The formulas in Theorems 1 and  2 can be implemented in Mathematica as follows:

  NonCentral[m_, x_, p_, d_] :=

    Sum[FactorialPower[m, Sum[k[i], {i,1,d}]] *

    Product[StirlingS2[p[[i]], k[i]] * x[[i]] ^ k[i], {i,1,d}], ##] & @@

    ({k[#], 0, p[[#]]} & /@ Range[d]);

  Central[m_, x_, p_, d_] :=

    Sum[Sum[FactorialPower[m, Sum[k[i], {i,1,d}]] *

    (-m) ^ Sum[p[[i]] - ell[i], {i,1,d}] *

    Product[Binomial[p[[i]], ell[i]] * StirlingS2[ell[i], k[i]] *

    x[[i]] ^ (p[[i]] - ell[i] + k[i]), {i,1,d}], ##] & @@

    ({k[#], 0, ell[#]} & /@ Range[d]), ##] & @@

    ({ell[#], 0, p[[#]]} & /@ Range[d]);

5. Explicit Formulas

In Newcomer (2008) [2], explicit expressions for the non-central moments of order 3 and 4 were obtained for the multinomial distribution, see also Newcomer et al. (2008) and Ouimet (2020) [43,44]. 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:

$$\begin{array}{cc}\hfill & \left\{\genfrac{}{}{0.0pt}{}{0}{0}\right\}=1,\hfill \\ \hfill & \left\{\genfrac{}{}{0.0pt}{}{1}{0}\right\}=0,\left\{\genfrac{}{}{0.0pt}{}{1}{1}\right\}=1,\hfill \\ \hfill & \left\{\genfrac{}{}{0.0pt}{}{2}{0}\right\}=0,\left\{\genfrac{}{}{0.0pt}{}{2}{1}\right\}=1,\left\{\genfrac{}{}{0.0pt}{}{2}{2}\right\}=1,\hfill \\ \hfill & \left\{\genfrac{}{}{0.0pt}{}{3}{0}\right\}=0,\left\{\genfrac{}{}{0.0pt}{}{3}{1}\right\}=1,\left\{\genfrac{}{}{0.0pt}{}{3}{2}\right\}=3,\left\{\genfrac{}{}{0.0pt}{}{3}{3}\right\}=1,\hfill \\ \hfill & \left\{\genfrac{}{}{0.0pt}{}{4}{0}\right\}=0,\left\{\genfrac{}{}{0.0pt}{}{4}{1}\right\}=1,\left\{\genfrac{}{}{0.0pt}{}{4}{2}\right\}=7,\left\{\genfrac{}{}{0.0pt}{}{4}{3}\right\}=6,\left\{\genfrac{}{}{0.0pt}{}{4}{4}\right\}=1,\hfill \\ \hfill & \left\{\genfrac{}{}{0.0pt}{}{5}{0}\right\}=0,\left\{\genfrac{}{}{0.0pt}{}{5}{1}\right\}=1,\left\{\genfrac{}{}{0.0pt}{}{5}{2}\right\}=15,\left\{\genfrac{}{}{0.0pt}{}{5}{3}\right\}=25,\left\{\genfrac{}{}{0.0pt}{}{5}{4}\right\}=10,\left\{\genfrac{}{}{0.0pt}{}{5}{5}\right\}=1,\hfill \\ \hfill & \left\{\genfrac{}{}{0.0pt}{}{6}{0}\right\}=0,\left\{\genfrac{}{}{0.0pt}{}{6}{1}\right\}=1,\left\{\genfrac{}{}{0.0pt}{}{6}{2}\right\}=31,\left\{\genfrac{}{}{0.0pt}{}{6}{3}\right\}=90,\left\{\genfrac{}{}{0.0pt}{}{6}{4}\right\}=65,\left\{\genfrac{}{}{0.0pt}{}{6}{5}\right\}=15,\left\{\genfrac{}{}{0.0pt}{}{6}{6}\right\}=1,\hfill \\ \hfill & \left\{\genfrac{}{}{0.0pt}{}{7}{0}\right\}=0,\left\{\genfrac{}{}{0.0pt}{}{7}{1}\right\}=1,\left\{\genfrac{}{}{0.0pt}{}{7}{2}\right\}=63,\left\{\genfrac{}{}{0.0pt}{}{7}{3}\right\}=301,\left\{\genfrac{}{}{0.0pt}{}{7}{4}\right\}=350,\left\{\genfrac{}{}{0.0pt}{}{7}{5}\right\}=140,\left\{\genfrac{}{}{0.0pt}{}{7}{6}\right\}=21,\left\{\genfrac{}{}{0.0pt}{}{7}{7}\right\}=1,\hfill \\ \hfill & \left\{\genfrac{}{}{0.0pt}{}{8}{0}\right\}=0,\left\{\genfrac{}{}{0.0pt}{}{8}{1}\right\}=1,\left\{\genfrac{}{}{0.0pt}{}{8}{2}\right\}=127,\left\{\genfrac{}{}{0.0pt}{}{8}{3}\right\}=966,\left\{\genfrac{}{}{0.0pt}{}{8}{4}\right\}=1701,\left\{\genfrac{}{}{0.0pt}{}{8}{5}\right\}=1050,\left\{\genfrac{}{}{0.0pt}{}{8}{6}\right\}=266,\left\{\genfrac{}{}{0.0pt}{}{8}{7}\right\}=28,\left\{\genfrac{}{}{0.0pt}{}{8}{8}\right\}=1.\hfill \end{array}$$

5.1. Computation of the Non-Central Moments Up to Order 8

By applying the general expression in Theorem 1 and by removing the Stirling numbers $\left\{\genfrac{}{}{0.0pt}{}{p_i}{k_i}\right\}$ that are equal to 0, we get the following results directly.

Order 1: For any $j_1\in \{1,2,\dots ,d\}$,

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}]& =x_{j_1}m.\hfill \end{array}$$

(11)

Order 2: For any distinct $j_1,j_2\in \{1,2,\dots ,d\}$,

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^2]& =x_{j_1}[m+m^{(2)}{x}_{j_1}],\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}{\xi }_{j_2}]& =x_{j_1}{x}_{j_2}{m}^{(2)}.\hfill \end{array}$$

(13)

Order 3: For any distinct $j_1,j_2,j_3\in \{1,2,\dots ,d\}$,

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^3]& =x_{j_1}[m+3m^{(2)}{x}_{j_1}+m^{(3)}{x}_{j_1}^2],\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^2{\xi }_{j_2}]& =x_{j_1}{x}_{j_2}[m^{(2)}+m^{(3)}{x}_{j_1}],\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}{\xi }_{j_2}{\xi }_{j_3}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{m}^{(3)}.\hfill \end{array}$$

(16)

Order 4: For any distinct $j_1,j_2,j_3,j_4\in \{1,2,\dots ,d\}$,

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^4]& =x_{j_1}[m+7m^{(2)}{x}_{j_1}+6m^{(3)}{x}_{j_1}^2+m^{(4)}{x}_{j_1}^3],\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^3{\xi }_{j_2}]& =x_{j_1}{x}_{j_2}[m^{(2)}+3m^{(3)}{x}_{j_1}+m^{(4)}{x}_{j_1}^2],\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^2{\xi }_{j_2}^2]& =x_{j_1}{x}_{j_2}[m^{(2)}+m^{(3)}(x_{j_1}+x_{j_2})+m^{(4)}{x}_{j_1}{x}_{j_2}],\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^2{\xi }_{j_2}{\xi }_{j_3}]& =x_{j_1}{x}_{j_2}{x}_{j_3}]m^{(3)}+m^{(4)}{x}_{j_1}],\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}{\xi }_{j_2}{\xi }_{j_3}{\xi }_{j_4}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}{m}^{(4)}.\hfill \end{array}$$

(21)

Order 5: For any distinct $j_1,j_2,j_3,j_4,j_5\in \{1,2,\dots ,d\}$,

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^5]& =x_{j_1}[m+15m^{(2)}{x}_{j_1}+25m^{(3)}{x}_{j_1}^2+10m^{(4)}{x}_{j_1}^3+m^{(5)}{x}_{j_1}^4],\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^4{\xi }_{j_2}]& =x_{j_1}{x}_{j_2}[m^{(2)}+7m^{(3)}{x}_{j_1}+6m^{(4)}{x}_{j_1}^2+m^{(5)}{x}_{j_1}^3],\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^3{\xi }_{j_2}^2]& =x_{j_1}{x}_{j_2}[m^{(2)}+m^{(3)}(3x_{j_1}+x_{j_2})+m^{(4)}(x_{j_1}^2+3x_{j_1}{x}_{j_2})+m^{(5)}{x}_{j_1}^2{x}_{j_2}],\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^3{\xi }_{j_2}{\xi }_{j_3}]& =x_{j_1}{x}_{j_2}{x}_{j_3}[m^{(3)}+3m^{(4)}{x}_{j_1}+m^{(5)}{x}_{j_1}^2],\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^2{\xi }_{j_2}^2{\xi }_{j_3}]& =x_{j_1}{x}_{j_2}{x}_{j_3}[m^{(3)}+m^{(4)}(x_{j_1}+x_{j_2})+m^{(5)}{x}_{j_1}{x}_{j_2}],\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^2{\xi }_{j_2}{\xi }_{j_3}{\xi }_{j_4}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}[m^{(4)}+m^{(5)}{x}_{j_1}],\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}{\xi }_{j_2}{\xi }_{j_3}{\xi }_{j_4}{\xi }_{j_5}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}{x}_{j_5}{m}^{(5)}.\hfill \end{array}$$

(28)

Order 6: For any distinct $j_1,j_2,j_3,j_4,j_5,j_6\in \{1,2,\dots ,d\}$,

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^6]& =x_{j_1}[m+31m^{(2)}{x}_{j_1}+90m^{(3)}{x}_{j_1}^2+65m^{(4)}{x}_{j_1}^3+15m^{(5)}{x}_{j_1}^4+m^{(6)}{x}_{j_1}^5],\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^5{\xi }_{j_2}]& =x_{j_1}{x}_{j_2}[m^{(2)}+15m^{(3)}{x}_{j_1}+25m^{(4)}{x}_{j_1}^2+10m^{(5)}{x}_{j_1}^3+m^{(6)}{x}_{j_1}^4],\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^4{\xi }_{j_2}^2]& =x_{j_1}{x}_{j_2}\left[\begin{array}{c}{m}^{(2)}+m^{(3)}(7x_{j_1}+x_{j_2})+m^{(4)}(6x_{j_1}^2+7x_{j_1}{x}_{j_2})\hfill \\ +m^{(5)}(x_{j_1}^3+6x_{j_1}^2{x}_{j_2})+m^{(6)}{x}_{j_1}^3{x}_{j_2}\hfill \end{array}\right],\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^4{\xi }_{j_2}{\xi }_{j_3}]& =x_{j_1}{x}_{j_2}{x}_{j_3}[m^{(3)}+7m^{(4)}{x}_{j_1}+6m^{(5)}{x}_{j_1}^2+m^{(6)}{x}_{j_1}^3],\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^3{\xi }_{j_2}^3]& =x_{j_1}{x}_{j_2}\left[\begin{array}{c}{m}^{(2)}+m^{(3)}(3x_{j_1}+3x_{j_2})+m^{(4)}(x_{j_1}^2+9x_{j_1}{x}_{j_2}+x_{j_2}^2)\hfill \\ +m^{(5)}(3x_{j_1}^2{x}_{j_2}+3x_{j_1}{x}_{j_2}^2)+m^{(6)}{x}_{j_1}^2{x}_{j_2}^2\hfill \end{array}\right],\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^3{\xi }_{j_2}^2{\xi }_{j_3}]& =x_{j_1}{x}_{j_2}{x}_{j_3}[m^{(3)}+m^{(4)}(3x_{j_1}+x_{j_2})+m^{(5)}(x_{j_1}^2+3x_{j_1}{x}_{j_2})+m^{(6)}{x}_{j_1}^2{x}_{j_2}],\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^3{\xi }_{j_2}{\xi }_{j_3}{\xi }_{j_4}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}[m^{(4)}+3m^{(5)}{x}_{j_1}+m^{(6)}{x}_{j_1}^2],\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^2{\xi }_{j_2}^2{\xi }_{j_3}^2]& =x_{j_1}{x}_{j_2}{x}_{j_3}\left[\begin{array}{c}{m}^{(3)}+m^{(4)}(x_{j_1}+x_{j_2}+x_{j_3})\hfill \\ +m^{(5)}(x_{j_1}{x}_{j_2}+x_{j_1}{x}_{j_3}+x_{j_2}{x}_{j_3})+m^{(6)}{x}_{j_1}{x}_{j_2}{x}_{j_3}\hfill \end{array}\right],\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^2{\xi }_{j_2}^2{\xi }_{j_3}{\xi }_{j_4}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}[m^{(4)}+m^{(5)}(x_{j_1}+x_{j_2})+m^{(6)}{x}_{j_1}{x}_{j_2}],\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^2{\xi }_{j_2}{\xi }_{j_3}{\xi }_{j_4}{\xi }_{j_5}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}{x}_{j_5}[m^{(5)}+m^{(6)}{x}_{j_1}],\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}{\xi }_{j_2}{\xi }_{j_3}{\xi }_{j_4}{\xi }_{j_5}{\xi }_{j_6}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}{x}_{j_5}{x}_{j_6}{m}^{(6)}.\hfill \end{array}$$

(39)

Order 7: For any distinct $j_1,j_2,j_3,j_4,j_5,j_6,j_7\in \{1,2,\dots ,d\}$,

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^7]& =x_{j_1}\left[\begin{array}{c}m+63m^{(2)}{x}_{j_1}+301m^{(3)}{x}_{j_1}^2+350m^{(4)}{x}_{j_1}^3\hfill \\ +140m^{(5)}{x}_{j_1}^4+21m^{(6)}{x}_{j_1}^5+m^{(7)}{x}_{j_1}^6\hfill \end{array}\right],\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^6{\xi }_{j_2}]& =x_{j_1}{x}_{j_2}\left[\begin{array}{c}{m}^{(2)}+31m^{(3)}{x}_{j_1}+90m^{(4)}{x}_{j_1}^2\hfill \\ +65m^{(5)}{x}_{j_1}^3+15m^{(6)}{x}_{j_1}^4+m^{(7)}{x}_{j_1}^5\hfill \end{array}\right],\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^5{\xi }_{j_2}^2]& =x_{j_1}{x}_{j_2}\left[\begin{array}{c}{m}^{(2)}+m^{(3)}(15x_{j_1}+x_{j_2})+m^{(4)}(25x_{j_1}^2+15x_{j_1}{x}_{j_2})\hfill \\ +m^{(5)}(10x_{j_1}^3+25x_{j_1}^2{x}_{j_2})+m^{(6)}(x_{j_1}^4+10x_{j_1}^3{x}_{j_2})+m^{(7)}{x}_{j_1}^4{x}_{j_2}\hfill \end{array}\right],\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^5{\xi }_{j_2}{\xi }_{j_3}]& =x_{j_1}{x}_{j_2}{x}_{j_3}[m^{(3)}+15m^{(4)}{x}_{j_1}+25m^{(5)}{x}_{j_1}^2+10m^{(6)}{x}_{j_1}^3+m^{(7)}{x}_{j_1}^4],\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^4{\xi }_{j_2}^3]& =x_{j_1}{x}_{j_2}\left[\begin{array}{c}{m}^{(2)}+m^{(3)}(7x_{j_1}+3x_{j_2})+m^{(4)}(6x_{j_1}^2+21x_{j_1}{x}_{j_2}+x_{j_2}^2)\hfill \\ +m^{(5)}(x_{j_1}^3+18x_{j_1}^2{x}_{j_2}+7x_{j_1}{x}_{j_2}^2)\hfill \\ +m^{(6)}(3x_{j_1}^3{x}_{j_2}+6x_{j_1}^2{x}_{j_2}^2)+m^{(7)}{x}_{j_1}^3{x}_{j_2}^2\hfill \end{array}\right],\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^4{\xi }_{j_2}^2{\xi }_{j_3}]& =x_{j_1}{x}_{j_2}{x}_{j_3}\left[\begin{array}{c}{m}^{(3)}+m^{(4)}(7x_{j_1}+x_{j_2})+m^{(5)}(6x_{j_1}^2+7x_{j_1}{x}_{j_2})\hfill \\ +m^{(6)}(x_{j_1}^3+6x_{j_1}^2{x}_{j_2})+m^{(7)}{x}_{j_1}^3{x}_{j_2}\hfill \end{array}\right],\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^4{\xi }_{j_2}{\xi }_{j_3}{\xi }_{j_4}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}[m^{(4)}+7m^{(5)}{x}_{j_1}+6m^{(6)}{x}_{j_1}^2+m^{(7)}{x}_{j_1}^3],\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^3{\xi }_{j_2}^3{\xi }_{j_3}]& =x_{j_1}{x}_{j_2}{x}_{j_3}\left[\begin{array}{c}{m}^{(3)}+m^{(4)}(3x_{j_1}+3x_{j_2})+m^{(5)}(x_{j_1}^2+9x_{j_1}{x}_{j_2}+x_{j_2}^2)\hfill \\ +m^{(6)}(3x_{j_1}^2{x}_{j_2}+3x_{j_1}{x}_{j_2}^2)+m^{(7)}{x}_{j_1}^2{x}_{j_2}^2\hfill \end{array}\right],\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^3{\xi }_{j_2}^2{\xi }_{j_3}^2]& =x_{j_1}{x}_{j_2}{x}_{j_3}\left[\begin{array}{c}{m}^{(3)}+m^{(4)}(3x_{j_1}+x_{j_2}+x_{j_3})\hfill \\ +m^{(5)}(x_{j_1}^2+3x_{j_1}{x}_{j_2}+3x_{j_1}{x}_{j_3}+x_{j_2}{x}_{j_3})\hfill \\ +m^{(6)}(x_{j_1}^2{x}_{j_2}+x_{j_1}^2{x}_{j_3}+3x_{j_1}{x}_{j_2}{x}_{j_3})+m^{(7)}{x}_{j_1}^2{x}_{j_2}{x}_{j_3}\hfill \end{array}\right],\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^3{\xi }_{j_2}^2{\xi }_{j_3}{\xi }_{j_4}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}\left[\begin{array}{c}{m}^{(4)}+m^{(5)}(3x_{j_1}+x_{j_2})\hfill \\ +m^{(6)}(x_{j_1}^2+3x_{j_1}{x}_{j_2})+m^{(7)}{x}_{j_1}^2{x}_{j_2}\hfill \end{array}\right],\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^3{\xi }_{j_2}{\xi }_{j_3}{\xi }_{j_4}{\xi }_{j_5}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}{x}_{j_5}[m^{(5)}+3m^{(6)}{x}_{j_1}+m^{(7)}{x}_{j_1}^2],\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^2{\xi }_{j_2}^2{\xi }_{j_3}^2{\xi }_{j_4}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}\left[\begin{array}{c}{m}^{(4)}+m^{(5)}(x_{j_1}+x_{j_2}+x_{j_3})\hfill \\ +m^{(6)}(x_{j_1}{x}_{j_2}+x_{j_1}{x}_{j_3}+x_{j_2}{x}_{j_3})+m^{(7)}{x}_{j_1}{x}_{j_2}{x}_{j_3}\hfill \end{array}\right],\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^2{\xi }_{j_2}^2{\xi }_{j_3}{\xi }_{j_4}{\xi }_{j_5}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}{x}_{j_5}\left[m^{(5)}+m^{(6)}(x_{j_1}+x_{j_2})+m^{(7)}{x}_{j_1}{x}_{j_2}\right],\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^2{\xi }_{j_2}{\xi }_{j_3}{\xi }_{j_4}{\xi }_{j_5}{\xi }_{j_6}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}{x}_{j_5}{x}_{j_6}[m^{(6)}+m^{(7)}{x}_{j_1}],\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}{\xi }_{j_2}{\xi }_{j_3}{\xi }_{j_4}{\xi }_{j_5}{\xi }_{j_6}{\xi }_{j_7}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}{x}_{j_5}{x}_{j_6}{x}_{j_7}{m}^{(7)}.\hfill \end{array}$$

(54)

Order 8: For any distinct $j_1,j_2,j_3,j_4,j_5,j_6,j_7,j_8\in \{1,2,\dots ,d\}$,

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^8]& =x_{j_1}\left[\begin{array}{c}m+127m^{(2)}{x}_{j_1}+966m^{(3)}{x}_{j_2}^2+1701m^{(4)}{x}_{j_1}^3\hfill \\ +1050m^{(5)}{x}_{j_1}^4+266m^{(6)}{x}_{j_1}^5+28m^{(7)}{x}_{j_1}^6+m^{(8)}{x}_{j_1}^7\hfill \end{array}\right],\hfill \end{array}$$

(55)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^7{\xi }_{j_2}]& =x_{j_1}{x}_{j_2}\left[\begin{array}{c}m+63m^{(3)}{x}_{j_1}+301m^{(4)}{x}_{j_1}^2+350m^{(5)}{x}_{j_1}^3\hfill \\ +140m^{(6)}{x}_{j_1}^4+21m^{(7)}{x}_{j_1}^5+m^{(8)}{x}_{j_1}^6\hfill \end{array}\right],\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^6{\xi }_{j_2}^2]& =x_{j_1}{x}_{j_2}\left[\begin{array}{c}{m}^{(2)}+m^{(3)}(31x_{j_1}+x_{j_2})+m^{(4)}(90x_{j_1}^2+31x_{j_1}{x}_{j_2})\hfill \\ +m^{(5)}(65x_{j_1}^3+90x_{j_1}^2{x}_{j_2})+m^{(6)}(15x_{j_1}^4+65x_{j_1}^3{x}_{j_2})\hfill \\ +m^{(7)}(x_{j_1}^5+15x_{j_1}^4{x}_{j_2})+m^{(8)}{x}_{j_1}^5{x}_{j_2}\hfill \end{array}\right],\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^6{\xi }_{j_2}{\xi }_{j_3}]& =x_{j_1}{x}_{j_2}{x}_{j_3}\left[\begin{array}{c}{m}^{(3)}+31m^{(4)}{x}_{j_1}+90m^{(5)}{x}_{j_1}^2\hfill \\ +65m^{(6)}{x}_{j_1}^3+15m^{(7)}{x}_{j_1}^4+m^{(8)}{x}_{j_1}^5\hfill \end{array}\right],\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^5{\xi }_{j_2}^3]& =x_{j_1}{x}_{j_2}\left[\begin{array}{c}{m}^{(2)}+m^{(3)}(15x_{j_1}+3x_{j_2})+m^{(4)}(25x_{j_1}^2+45x_{j_1}{x}_{j_2}+x_{j_2}^2)\hfill \\ +m^{(5)}(10x_{j_1}^3+75x_{j_1}^2{x}_{j_2}+15x_{j_1}{x}_{j_2}^2)\hfill \\ +m^{(6)}(x_{j_4}^4+30x_{j_1}^3{x}_{j_2}+25x_{j_1}^2{x}_{j_2}^2)\hfill \\ +m^{(7)}(3x_{j_1}^4{x}_{j_2}+10x_{j_1}^3{x}_{j_2}^2)+m^{(8)}{x}_{j_1}^4{x}_{j_2}^2\hfill \end{array}\right],\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^5{\xi }_{j_2}^2{\xi }_{j_3}]& =x_{j_1}{x}_{j_2}{x}_{j_3}\left[\begin{array}{c}{m}^{(3)}+m^{(4)}(15x_{j_1}+x_{j_2})+m^{(5)}(25x_{j_1}^2+15x_{j_1}{x}_{j_2})\hfill \\ +m^{(6)}(10x_{j_1}^3+25x_{j_1}^2{x}_{j_2})+m^{(7)}(x_{j_1}^4+10x_{j_1}^3{x}_{j_2})+m^{(8)}{x}_{j_1}^4{x}_{j_2}\hfill \end{array}\right],\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^5{\xi }_{j_2}{\xi }_{j_3}{\xi }_{j_4}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}[m^{(4)}+15m^{(5)}{x}_{j_1}+25m^{(6)}{x}_{j_1}^2+10m^{(7)}{x}_{j_1}^3+m^{(8)}{x}_{j_1}^4],\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^4{\xi }_{j_2}^4]& =x_{j_1}{x}_{j_2}\left[\begin{array}{c}{m}^{(2)}+m^{(3)}(7x_{j_1}+7x_{j_2})+m^{(4)}(6x_{j_1}^2+49x_{j_1}{x}_{j_2}+6x_{j_2}^2)\hfill \\ +m^{(5)}(x_{j_1}^3+42x_{j_1}^2{x}_{j_2}+42x_{j_1}{x}_{j_2}^2+x_{j_2}^3)\hfill \\ +m^{(6)}(7x_{j_1}^3{x}_{j_2}+36x_{j_1}^2{x}_{j_2}^2+7x_{j_1}{x}_{j_2}^3)\hfill \\ +m^{(7)}(6x_{j_1}^3{x}_{j_2}^2+6x_{j_1}^2{x}_{j_2}^3)+m^{(8)}{x}_{j_1}^3{x}_{j_2}^3\hfill \end{array}\right],\hfill \end{array}$$

(62)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^4{\xi }_{j_2}^3{\xi }_{j_3}]& =x_{j_1}{x}_{j_2}{x}_{j_3}\left[\begin{array}{c}{m}^{(3)}+m^{(4)}(7x_{j_1}+3x_{j_2})+m^{(5)}(6x_{j_1}^2+21x_{j_1}{x}_{j_2}+x_{j_2}^2)\hfill \\ +m^{(6)}(x_{j_1}^3+18x_{j_1}^2{x}_{j_2}+7x_{j_1}{x}_{j_2}^2)\hfill \\ +m^{(7)}(3x_{j_1}^3{x}_{j_2}+6x_{j_1}^2{x}_{j_2}^2)+m^{(8)}{x}_{j_1}^3{x}_{j_2}^2\hfill \end{array}\right],\hfill \end{array}$$

(63)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^4{\xi }_{j_2}^2{\xi }_{j_3}^2]& =x_{j_1}{x}_{j_2}{x}_{j_3}\left[\begin{array}{c}{m}^{(3)}+m^{(4)}(7x_{j_1}+x_{j_2}+x_{j_3})\hfill \\ +m^{(5)}(6x_{j_1}^2+7x_{j_1}{x}_{j_2}+7x_{j_1}{x}_{j_3}+x_{j_2}{x}_{j_3})\hfill \\ +m^{(6)}(x_{j_1}^3+6x_{j_1}^2{x}_{j_2}+6x_{j_1}^2{x}_{j_3}+7x_{j_1}{x}_{j_2}{x}_{j_3})\hfill \\ +m^{(7)}(x_{j_1}^3{x}_{j_2}+x_{j_1}^3{x}_{j_3}+6x_{j_1}^2{x}_{j_2}{x}_{j_3})+m^{(8)}{x}_{j_1}^3{x}_{j_2}{x}_{j_3}\hfill \end{array}\right],\hfill \end{array}$$

(64)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^4{\xi }_{j_2}^2{\xi }_{j_3}{\xi }_{j_4}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}\left[\begin{array}{c}{m}^{(4)}+m^{(5)}(7x_{j_1}+x_{j_2})+m^{(6)}(6x_{j_1}^2+7x_{j_1}{x}_{j_2})\hfill \\ +m^{(7)}(x_{j_1}^3+6x_{j_1}^2{x}_{j_2})+m^{(8)}{x}_{j_1}^3{x}_{j_2}\hfill \end{array}\right],\hfill \end{array}$$

(65)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^4{\xi }_{j_2}{\xi }_{j_3}{\xi }_{j_4}{\xi }_{j_5}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}{x}_{j_5}[m^{(5)}+7m^{(6)}{x}_{j_1}+6m^{(7)}{x}_{j_1}^2+m^{(8)}{x}_{j_1}^3],\hfill \end{array}$$

(66)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^3{\xi }_{j_2}^3{\xi }_{j_3}^2]& =x_{j_1}{x}_{j_2}{x}_{j_3}\left[\begin{array}{c}{m}^{(3)}+m^{(4)}(3x_{j_1}+3x_{j_2}+x_{j_3})\hfill \\ +m^{(5)}(x_{j_1}^2+x_{j_2}^2+3x_{j_1}{x}_{j_3}+3x_{j_2}{x}_{j_3}+9x_{j_1}{x}_{j_2})\hfill \\ +m^{(6)}(x_{j_1}^2{x}_{j_3}+x_{j_2}^2{x}_{j_3}+3x_{j_1}^2{x}_{j_2}+3x_{j_1}{x}_{j_2}^2+9x_{j_1}{x}_{j_2}{x}_{j_3})\hfill \\ +m^{(7)}(x_{j_1}^2{x}_{j_2}^2+3x_{j_1}^2{x}_{j_2}{x}_{j_3}+3x_{j_1}{x}_{j_2}^2{x}_{j_3})+m^{(8)}{x}_{j_1}^2{x}_{j_2}^2{x}_{j_3}\hfill \end{array}\right],\hfill \end{array}$$

(67)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^3{\xi }_{j_2}^3{\xi }_{j_3}{\xi }_{j_4}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}\left[\begin{array}{c}{m}^{(4)}+m^{(5)}(3x_{j_1}+3x_{j_2})+m^{(6)}(x_{j_1}^2+9x_{j_1}{x}_{j_2}+x_{j_2}^2)\hfill \\ +m^{(7)}(3x_{j_1}^2{x}_{j_2}+3x_{j_1}{x}_{j_2}^2)+m^{(8)}{x}_{j_1}^2{x}_{j_2}^2\hfill \end{array}\right],\hfill \end{array}$$

(68)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^3{\xi }_{j_2}^2{\xi }_{j_3}^2{\xi }_{j_4}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}\left[\begin{array}{c}{m}^{(4)}+m^{(5)}(3x_{j_1}+x_{j_2}+x_{j_3})\hfill \\ +m^{(6)}(3x_{j_1}{x}_{j_2}+3x_{j_1}{x}_{j_3}+x_{j_2}{x}_{j_3})\hfill \\ +m^{(7)}(x_{j_1}^2{x}_{j_2}+x_{j_1}^2{x}_{j_3}+3x_{j_1}{x}_{j_2}{x}_{j_3})+m^{(8)}{x}_{j_1}^2{x}_{j_2}{x}_{j_3}\hfill \end{array}\right],\hfill \end{array}$$

(69)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^3{\xi }_{j_2}^2{\xi }_{j_3}{\xi }_{j_4}{\xi }_{j_5}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}{x}_{j_5}\left[\begin{array}{c}{m}^{(5)}+m^{(6)}(3x_{j_1}+x_{j_2})\hfill \\ +m^{(7)}(x_{j_1}^2+3x_{j_1}{x}_{j_2})+m^{(8)}{x}_{j_1}^2{x}_{j_2}\hfill \end{array}\right],\hfill \end{array}$$

(70)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^3{\xi }_{j_2}{\xi }_{j_3}{\xi }_{j_4}{\xi }_{j_5}{\xi }_{j_6}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}{x}_{j_5}{x}_{j_6}[m^{(6)}+3m^{(7)}{x}_{j_1}+m^{(8)}{x}_{j_1}^2],\hfill \end{array}$$

(71)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^2{\xi }_{j_2}^2{\xi }_{j_3}^2{\xi }_{j_4}^2]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}\left[\begin{array}{c}{m}^{(4)}+m^{(5)}(x_{j_1}+x_{j_2}+x_{j_3}+x_{j_4})\hfill \\ +m^{(6)}(x_{j_1}{x}_{j_2}+x_{j_1}{x}_{j_3}+x_{j_1}{x}_{j_4}+x_{j_2}{x}_{j_3}+x_{j_2}{x}_{j_4}+x_{j_3}{x}_{j_4})\hfill \\ +m^{(7)}(x_{j_1}{x}_{j_2}{x}_{j_3}+x_{j_1}{x}_{j_2}{x}_{j_4}+x_{j_1}{x}_{j_3}{x}_{j_4}+x_{j_2}{x}_{j_3}{x}_{j_4})\hfill \\ +m^{(8)}{x}_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}\hfill \end{array}\right],\hfill \end{array}$$

(72)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^2{\xi }_{j_2}^2{\xi }_{j_3}^2{\xi }_{j_4}{\xi }_{j_5}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}{x}_{j_5}\left[\begin{array}{c}{m}^{(5)}+m^{(6)}(x_{j_1}+x_{j_2}+x_{j_3})\hfill \\ +m^{(7)}(x_{j_1}{x}_{j_2}+x_{j_1}{x}_{j_3}+x_{j_2}{x}_{j_3})+m^{(8)}{x}_{j_1}{x}_{j_2}{x}_{j_3}\hfill \end{array}\right],\hfill \end{array}$$

(73)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^2{\xi }_{j_2}^2{\xi }_{j_3}{\xi }_{j_4}{\xi }_{j_5}{\xi }_{j_6}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}{x}_{j_5}{x}_{j_6}[m^{(6)}+m^{(7)}(x_{j_1}+x_{j_2})+m^{(8)}{x}_{j_1}{x}_{j_2}],\hfill \end{array}$$

(74)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}^2{\xi }_{j_2}{\xi }_{j_3}{\xi }_{j_4}{\xi }_{j_5}{\xi }_{j_6}{\xi }_{j_7}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}{x}_{j_5}{x}_{j_6}{x}_{j_7}[m^{(7)}+m^{(8)}{x}_{j_1}],\hfill \end{array}$$

(75)

$$\begin{array}{cc}\hfill \mathbb{E}[{\xi }_{j_1}{\xi }_{j_2}{\xi }_{j_3}{\xi }_{j_4}{\xi }_{j_5}{\xi }_{j_6}{\xi }_{j_7}{\xi }_{j_8}]& =x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}{x}_{j_5}{x}_{j_6}{x}_{j_7}{x}_{j_8}{m}^{(8)}.\hfill \end{array}$$

(76)

5.2. 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.

Order 2: For any distinct $j_1,j_2\in \{1,2,\dots ,d\}$,

$$\begin{array}{cc}\hfill \mathbb{E}[{({\xi }_{j_1}-\mathbb{E}[{\xi }_{j_1}])}^2]& =\mathbb{E}[{\xi }_{j_1}^2]-{(\mathbb{E}[{\xi }_{j_1}])}^2=x_{j_1}[m+m^{(2)}{x}_{j_1}]-m^2{x}_{j_1}^2\hfill \\ \hfill & =\begin{array}{|c|}\hline m{x}_{j_1}(1-x_{j_1})\\ \hline\end{array}\hfill \end{array}$$

(77)

$$\begin{array}{cc}\hfill \mathbb{E}[({\xi }_{j_1}-\mathbb{E}[{\xi }_{j_1}])({\xi }_{j_2}-\mathbb{E}[{\xi }_{j_2}])]& \hfill =\mathbb{E}[{\xi }_{j_1}{\xi }_{j_2}]-\mathbb{E}[{\xi }_{j_1}]\mathbb{E}[{\xi }_{j_2}]=m^{(2)}{x}_{j_1}{x}_{j_2}-m{x}_{j_1}m{x}_{j_2}\\ \hfill & =\begin{array}{|c|}\hline -m{x}_{j_1}{x}_{j_2}\\ \hline\end{array}.\hfill \end{array}$$

(78)

Order 3: For any distinct $j_1,j_2,j_3\in \{1,2,\dots ,d\}$,

$$\begin{array}{cc}\hfill \mathbb{E}[{({\xi }_{j_1}-\mathbb{E}[{\xi }_{j_1}])}^3]& =\mathbb{E}[{\xi }_{j_1}^3]-3\mathbb{E}[{\xi }_{j_1}^2]\mathbb{E}[{\xi }_{j_1}]+2{(\mathbb{E}[{\xi }_{j_1}])}^3\hfill \\ \hfill & =x_{j_1}[m+3m^{(2)}{x}_{j_1}+m^{(3)}{x}_{j_1}^2]-3x_{j_1}[m+m^{(2)}{x}_{j_1}]m{x}_{j_1}+2m^3{x}_{j_1}^3\hfill \\ \hfill & =\begin{array}{|c|}\hline m{x}_{j_1}(x_{j_1}-1)(2x_{j_1}-1)\\ \hline\end{array}\hfill \end{array}$$

(79)

$$\begin{array}{cc}\hfill & \mathbb{E}[{({\xi }_{j_1}-\mathbb{E}[{\xi }_{j_1}])}^2({\xi }_{j_2}-\mathbb{E}[{\xi }_{j_2}])]\hfill \\ \hfill & =\mathbb{E}[{\xi }_{j_1}^2{\xi }_{j_2}]-\mathbb{E}[{\xi }_{j_1}^2]\mathbb{E}[{\xi }_{j_2}]-2\mathbb{E}[{\xi }_{j_1}{\xi }_{j_2}]\mathbb{E}[{\xi }_{j_1}]+2{(\mathbb{E}[{\xi }_{j_1}])}^2\mathbb{E}[{\xi }_{j_2}]\hfill \\ \hfill & =x_{j_1}{x}_{j_2}[m^{(2)}+m^{(3)}{x}_{j_1}]-x_{j_1}\left[m+m^{(2)}{x}_{j_1}\right]m{x}_{j_2}-2m^{(2)}{x}_{j_1}{x}_{j_2}m{x}_{j_1}+2m^2{x}_{j_1}^{2}m{x}_{j_2}\hfill \\ \hfill & =\begin{array}{|c|}\hline m{x}_{j_1}{x}_{j_2}(2x_{j_1}-1)\\ \hline\end{array}\hfill \end{array}$$

(80)

$$\begin{array}{cc}\hfill & \mathbb{E}[({\xi }_{j_1}-\mathbb{E}[{\xi }_{j_1}])({\xi }_{j_2}-\mathbb{E}[{\xi }_{j_2}])({\xi }_{j_3}-\mathbb{E}[{\xi }_{j_3}])]\hfill \\ \hfill & =\mathbb{E}[{\xi }_{j_1}{\xi }_{j_2}{\xi }_{j_3}]-\mathbb{E}[{\xi }_{j_1}{\xi }_{j_2}]\mathbb{E}[{\xi }_{j_3}]-\mathbb{E}[{\xi }_{j_1}{\xi }_{j_3}]\mathbb{E}[{\xi }_{j_2}]-\mathbb{E}[{\xi }_{j_2}{\xi }_{j_3}]\mathbb{E}[{\xi }_{j_1}]+2\mathbb{E}[{\xi }_{j_1}]\mathbb{E}[{\xi }_{j_2}]\mathbb{E}[{\xi }_{j_3}]\hfill \\ \hfill & =m^{(3)}{x}_{j_1}{x}_{j_2}{x}_{j_3}-m^{(2)}{x}_{j_1}{x}_{j_2}m{x}_{j_3}-m^{(2)}{x}_{j_1}{x}_{j_3}m{x}_{j_2}-m^{(2)}{x}_{j_2}{x}_{j_3}m{x}_{j_1}+2m^3{x}_{j_1}{x}_{j_2}{x}_{j_3}\hfill \\ \hfill & =\begin{array}{|c|}\hline 2m{x}_{j_1}{x}_{j_2}{x}_{j_3}\\ \hline\end{array}.\hfill \end{array}$$

(81)

$$\begin{array}{cc}\hfill & \mathbb{E}[{({\xi }_{j_1}-\mathbb{E}[{\xi }_{j_1}])}^4]\hfill \\ \hfill & =\mathbb{E}[{\xi }_{j_1}^4]-4\mathbb{E}[{\xi }_{j_1}^3]\mathbb{E}[{\xi }_{j_1}]+6\mathbb{E}[{\xi }_{j_1}^2]{(\mathbb{E}[{\xi }_{j_1}])}^2-3{(\mathbb{E}[{\xi }_{j_1}])}^4\hfill \\ \hfill & =x_{j_1}[m+7m^{(2)}{x}_{j_1}+6m^{(3)}{x}_{j_1}^2+m^{(4)}{x}_{j_1}^3]-4x_{j_1}[m+3m^{(2)}{x}_{j_1}+m^{(3)}{x}_{j_1}^2]m{x}_{j_1}\hfill \\ \hfill & +6x_{j_1}[m+m^{(2)}{x}_{j_1}]{(m{x}_{j_1})}^2-3m^4{x}_{j_1}^4\hfill \\ \hfill & =\begin{array}{|c|}\hline 3m^2{x}_{j_1}^2{(x_{j_1}-1)}^2+m{x}_{j_1}(1-x_{j_1})(6x_{j_1}^2-6x_{j_1}+1)\\ \hline\end{array}\hfill \end{array}$$

(82)

Order 4: For any distinct $j_1,j_2,j_3,j_4\in \{1,2,\dots ,d\}$,

$$\begin{array}{cc}\hfill & \mathbb{E}[{({\xi }_{j_1}-\mathbb{E}[{\xi }_{j_1}])}^3({\xi }_{j_2}-\mathbb{E}[{\xi }_{j_2}])]\hfill \\ \hfill & =\mathbb{E}[{\xi }_{j_1}^3{\xi }_{j_2}]-\mathbb{E}[{\xi }_{j_1}^3]\mathbb{E}[{\xi }_{j_2}]-3\mathbb{E}[{\xi }_{j_1}^2{\xi }_{j_2}]\mathbb{E}[{\xi }_{j_1}]+3\mathbb{E}[{\xi }_{j_1}^2]\mathbb{E}[{\xi }_{j_1}]\mathbb{E}[{\xi }_{j_2}]\hfill \\ \hfill & +3\mathbb{E}[{\xi }_{j_1}{\xi }_{j_2}]{(\mathbb{E}[{\xi }_{j_1}])}^2-3{(\mathbb{E}[{\xi }_{j_1}])}^3\mathbb{E}[{\xi }_{j_2}]\hfill \\ \hfill & =x_{j_1}{x}_{j_2}[m^{(2)}+3m^{(3)}{x}_{j_1}+m^{(4)}{x}_{j_1}^2]-x_{j_1}[m+3m^{(2)}{x}_{j_1}+m^{(3)}{x}_{j_1}^2]m{x}_{j_2}\hfill \\ \hfill & -3x_{j_1}{x}_{j_2}[m^{(2)}+m^{(3)}{x}_{j_1}]m{x}_{j_1}+3x_{j_1}[m+m^{(2)}{x}_{j_1}]m{x}_{j_1}m{x}_{j_2}+3m^{(2)}{x}_{j_1}{x}_{j_2}{m}^2{x}_{j_1}^2-3m^3{x}_{j_1}^{3}m{x}_{j_2}\hfill \\ \hfill & =\begin{array}{|c|}\hline m{x}_{j_1}{x}_{j_2}(3(m-2)x_{j_1}(x_{j_1}-1)-1)\\ \hline\end{array}\hfill \end{array}$$

(83)

$$\begin{array}{cc}\hfill & \mathbb{E}[{({\xi }_{j_1}-\mathbb{E}[{\xi }_{j_1}])}^2{({\xi }_{j_2}-\mathbb{E}[{\xi }_{j_2}])}^2]\hfill \\ \hfill & =\mathbb{E}[{\xi }_{j_1}^2{\xi }_{j_2}^2]-2\mathbb{E}[{\xi }_{j_1}^2{\xi }_{j_2}]\mathbb{E}[{\xi }_{j_2}]-2\mathbb{E}[{\xi }_{j_1}{\xi }_{j_2}^2]\mathbb{E}[{\xi }_{j_1}]+\mathbb{E}[{\xi }_{j_1}^2]{(\mathbb{E}[{\xi }_{j_2}])}^2+\mathbb{E}[{\xi }_{j_2}^2]{(\mathbb{E}[{\xi }_{j_1}])}^2\hfill \\ \hfill & +4\mathbb{E}[{\xi }_{j_1}{\xi }_{j_2}]\mathbb{E}[{\xi }_{j_1}]\mathbb{E}[{\xi }_{j_2}]-3{(\mathbb{E}[{\xi }_{j_1}])}^2{(\mathbb{E}[{\xi }_{j_2}])}^2\hfill \\ \hfill & =x_{j_1}{x}_{j_2}[m^{(2)}+m^{(3)}(x_{j_1}+x_{j_2})+m^{(4)}{x}_{j_1}{x}_{j_2}]-2x_{j_1}{x}_{j_2}[m^{(2)}+m^{(3)}{x}_{j_1}]m{x}_{j_2}\hfill \\ \hfill & -2x_{j_1}{x}_{j_2}[m^{(2)}+m^{(3)}{x}_{j_2}]m{x}_{j_1}+x_{j_1}[m+m^{(2)}{x}_{j_1}]m^2{x}_{j_2}^2+x_{j_2}[m+m^{(2)}{x}_{j_2}]m^2{x}_{j_1}^2\hfill \\ \hfill & +4m^{(2)}{x}_{j_1}{x}_{j_2}m{x}_{j_1}m{x}_{j_2}-3m^2{x}_{j_1}^2{m}^2{x}_{j_2}^2\hfill \\ \hfill & =\begin{array}{|c|}\hline m(m-2)x_{j_1}{x}_{j_2}(3x_{j_1}{x}_{j_2}-(x_{j_1}+x_{j_2})+1)+m{x}_{j_1}{x}_{j_2}\\ \hline\end{array}\hfill \end{array}$$

(84)

$$\begin{array}{cc}\hfill & \mathbb{E}[{({\xi }_{j_1}-\mathbb{E}[{\xi }_{j_1}])}^2({\xi }_{j_2}-\mathbb{E}[{\xi }_{j_2}])({\xi }_{j_3}-\mathbb{E}[{\xi }_{j_3}])]\hfill \\ \hfill & =\mathbb{E}[{\xi }_{j_1}^2{\xi }_{j_2}{\xi }_{j_3}]-\mathbb{E}[{\xi }_{j_1}^2{\xi }_{j_2}]\mathbb{E}[{\xi }_{j_3}]-\mathbb{E}[{\xi }_{j_1}^2{\xi }_{j_3}]\mathbb{E}[{\xi }_{j_2}]-2\mathbb{E}[{\xi }_{j_1}{\xi }_{j_2}{\xi }_{j_3}]\mathbb{E}[{\xi }_{j_1}]+\mathbb{E}[{\xi }_{j_1}^2]\mathbb{E}[{\xi }_{j_2}]\mathbb{E}[{\xi }_{j_3}]\hfill \\ \hfill & +2\mathbb{E}[{\xi }_{j_1}{\xi }_{j_2}]\mathbb{E}[{\xi }_{j_1}]\mathbb{E}[{\xi }_{j_3}]+2\mathbb{E}[{\xi }_{j_1}{\xi }_{j_3}]\mathbb{E}[{\xi }_{j_1}]\mathbb{E}[{\xi }_{j_2}]+\mathbb{E}[{\xi }_{j_2}{\xi }_{j_3}]{(\mathbb{E}[{\xi }_{j_1}])}^2-3{(\mathbb{E}[{\xi }_{j_1}])}^2\mathbb{E}[{\xi }_{j_2}]\mathbb{E}[{\xi }_{j_3}]\hfill \\ \hfill & =x_{j_1}{x}_{j_2}{x}_{j_3}[m^{(3)}+m^{(4)}{x}_{j_1}]-x_{j_1}{x}_{j_2}[m^{(2)}+m^{(3)}{x}_{j_1}]m{x}_{j_3}-x_{j_1}{x}_{j_3}[m^{(2)}+m^{(3)}{x}_{j_1}]m{x}_{j_2}\hfill \\ \hfill & -2m^{(3)}{x}_{j_1}{x}_{j_2}{x}_{j_3}m{x}_{j_1}+x_{j_1}[m+m^{(2)}{x}_{j_1}]m{x}_{j_2}m{x}_{j_3}+2m^{(2)}{x}_{j_1}{x}_{j_2}m{x}_{j_1}m{x}_{j_3}\hfill \\ \hfill & +2m^{(2)}{x}_{j_1}{x}_{j_3}m{x}_{j_1}m{x}_{j_2}+m^{(2)}{x}_{j_2}{x}_{j_3}{m}^2{x}_{j_1}^2-3m^2{x}_{j_1}^{2}m{x}_{j_2}m{x}_{j_3}\hfill \\ \hfill & =\begin{array}{|c|}\hline m(m-2)x_{j_1}{x}_{j_2}{x}_{j_3}(3x_{j_1}-1)\\ \hline\end{array}\hfill \end{array}$$

(85)

$$\begin{array}{cc}\hfill & \mathbb{E}[({\xi }_{j_1}-\mathbb{E}[{\xi }_{j_1}])({\xi }_{j_2}-\mathbb{E}[{\xi }_{j_2}])({\xi }_{j_3}-\mathbb{E}[{\xi }_{j_3}])({\xi }_{j_4}-\mathbb{E}[{\xi }_{j_4}])]\hfill \\ \hfill & =\mathbb{E}[{\xi }_{j_1}{\xi }_{j_2}{\xi }_{j_3}{\xi }_{j_4}]-\mathbb{E}[{\xi }_{j_1}{\xi }_{j_2}{\xi }_{j_3}]\mathbb{E}[{\xi }_{j_4}]-\mathbb{E}[{\xi }_{j_1}{\xi }_{j_2}{\xi }_{j_4}]\mathbb{E}[{\xi }_{j_3}]-\mathbb{E}[{\xi }_{j_1}{\xi }_{j_3}{\xi }_{j_4}]\mathbb{E}[{\xi }_{j_2}]-\mathbb{E}[{\xi }_{j_2}{\xi }_{j_3}{\xi }_{j_4}]\mathbb{E}[{\xi }_{j_1}]\hfill \\ \hfill & +\mathbb{E}[{\xi }_{j_1}{\xi }_{j_2}]\mathbb{E}[{\xi }_{j_3}]\mathbb{E}[{\xi }_{j_4}]+\mathbb{E}[{\xi }_{j_1}{\xi }_{j_3}]\mathbb{E}[{\xi }_{j_2}]\mathbb{E}[{\xi }_{j_4}]+\mathbb{E}[{\xi }_{j_1}{\xi }_{j_4}]\mathbb{E}[{\xi }_{j_2}]\mathbb{E}[{\xi }_{j_3}]\hfill \\ \hfill & +\mathbb{E}[{\xi }_{j_2}{\xi }_{j_3}]\mathbb{E}[{\xi }_{j_1}]\mathbb{E}[{\xi }_{j_4}]+\mathbb{E}[{\xi }_{j_2}{\xi }_{j_4}]\mathbb{E}[{\xi }_{j_1}]\mathbb{E}[{\xi }_{j_3}]+\mathbb{E}[{\xi }_{j_3}{\xi }_{j_4}]\mathbb{E}[{\xi }_{j_1}]\mathbb{E}[{\xi }_{j_2}]-3\mathbb{E}[{\xi }_{j_1}]\mathbb{E}[{\xi }_{j_2}]\mathbb{E}[{\xi }_{j_3}]\mathbb{E}[{\xi }_{j_4}]\hfill \\ \hfill & =m^{(4)}{x}_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}-m^{(3)}{x}_{j_1}{x}_{j_2}{x}_{j_3}m{x}_{j_4}-m^{(3)}{x}_{j_1}{x}_{j_2}{x}_{j_4}m{x}_{j_3}-m^{(3)}{x}_{j_1}{x}_{j_3}{x}_{j_4}m{x}_{j_2}-m^{(3)}{x}_{j_2}{x}_{j_3}{x}_{j_4}m{x}_{j_1}\hfill \\ \hfill & +m^{(2)}{x}_{j_1}{x}_{j_2}m{x}_{j_3}m{x}_{j_4}+m^{(2)}{x}_{j_1}{x}_{j_3}m{x}_{j_2}m{x}_{j_4}+m^{(2)}{x}_{j_1}{x}_{j_4}m{x}_{j_2}m{x}_{j_3}\hfill \\ \hfill & +m^{(2)}{x}_{j_2}{x}_{j_3}m{x}_{j_1}m{x}_{j_4}+m^{(2)}{x}_{j_2}{x}_{j_4}m{x}_{j_1}m{x}_{j_3}+m^{(2)}{x}_{j_3}{x}_{j_4}m{x}_{j_1}m{x}_{j_2}-3m^4{x}_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}\hfill \\ \hfill & =\begin{array}{|c|}\hline 3m(m-2)x_{j_1}{x}_{j_2}{x}_{j_3}{x}_{j_4}\\ \hline\end{array}.\hfill \end{array}$$

(86)

6. Conclusions

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 [2], where the central moments were calculated up to order 4. It also complements the general formula for the (joint) factorial moments from [1] and the explicit formulas for some of the lower-order (mixed) cumulants that were presented in [3].