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

Ouimet, Frédéric

doi:10.3390/stats4010002

Open AccessArticle

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

by

Frédéric Ouimet

Department of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA

Stats 2021, 4(1), 18-27; https://doi.org/10.3390/stats4010002

Received: 10 November 2020 / Revised: 31 December 2020 / Accepted: 5 January 2021 / Published: 6 January 2021

(This article belongs to the Section Multivariate Analysis)

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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.

Keywords: multinomial distribution; higher moments; central moments; non-central moments

1. The Multinomial Distribution

For any

d \in N

, the d-dimensional (unit) simplex is defined by

S : = \{x \in {[0, 1]}^{d} : \sum_{i = 1}^{d} x_{i} \leq 1\}

, and the probability mass function

k \mapsto P_{k, m} (x)

for

ξ : = (ξ_{1}, ξ_{2}, \dots, ξ_{d}) \sim Multinomial (m, x)

is defined by

P_{k, m} (x) : = \frac{m!}{(m - \sum_{i = 1}^{d} k_{i})! \prod_{i = 1}^{d} k_{i}!} \cdot {(1 - \sum_{i = 1}^{d} x_{i})}^{m - \sum_{i = 1}^{d} k_{i}} \prod_{i = 1}^{d} x_{i}^{k_{i}}, k \in N_{0}^{d} \cap m S,

(1)

where

m \in N

and

x \in S

. If

x_{d + 1} : = 1 - \sum_{i = 1}^{d} x_{i}

, then (1) is just a reparametrization of

(ξ, 1 - \sum_{i = 1}^{d} ξ_{i}) \sim Multinomial (m, (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

E \prod_{i = 1}^{d} ξ_{i}^{p_{i}}] and E \prod_{i = 1}^{d} {(ξ_{i} - E [ξ_{i}])}^{p_{i}}], p_{1}, p_{2}, \dots, p_{d} \in 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

ξ \sim Multinomial (m, x)

. Then, for all

k_{1}, k_{2}, \dots, k_{d} \in N_{0}

,

E \prod_{i = 1}^{d} ξ_{i}^{(k_{i})}] = 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

X_{1}, X_{2}, \dots, X_{n}

, two examples of such estimators are the Bernstein estimator for the cumulative distribution function

F_{n, m}^{⋆} (x) : = \sum_{k \in N_{0}^{d} \cap m S} \frac{1}{n} \sum_{i = 1}^{n} 𝟙_{(- \infty, \frac{k}{m}]} (X_{i}) P_{k, m} (x), x \in S, m, n \in N,

(4)

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

{\hat{f}}_{n, m} (x) : = \sum_{k \in N_{0}^{d} \cap (m - 1) S} \frac{m^{d}}{n} \sum_{i = 1}^{n} 𝟙_{(\frac{k}{m}, \frac{k + 1}{m}]} (X_{i}) P_{k, m - 1} (x), x \in S, m, n \in 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 \geq 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 \geq 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

ξ \sim Multinomial (m, x)

. For all

p_{1}, p_{2}, \dots, p_{d} \in N_{0}

,

E \prod_{i = 1}^{d} ξ_{i}^{p_{i}}] = \sum_{k_{1} = 0}^{p_{1}} \dots \sum_{k_{d} = 0}^{p_{d}} m^{(\sum_{i = 1}^{d} k_{i})} \prod_{i = 1}^{d} \binom{p_{i}}{k_{i}}\} x_{i}^{k_{i}},

(6)

where

\{\binom{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 \in N_{0}

of a number

x \in R

and the falling factorials of x:

x^{p} = \sum_{k = 0}^{p} \binom{p}{k}\} x^{(k)} .

(7)

See, e.g., (Graham et al. (1994) [42], page 262). Apply this relation to every

ξ_{i}^{p_{i}}

and use the linearity of the expectation to get

E \prod_{i = 1}^{d} ξ_{i}^{p_{i}}] = \sum_{k_{1} = 0}^{p_{1}} \dots \sum_{k_{d} = 0}^{p_{d}} \binom{p_{1}}{k_{1}}\} \dots \binom{p_{d}}{k_{d}}\} E \prod_{i = 1}^{d} ξ_{i}^{(k_{i})}] .

(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

ξ \sim Multinomial (m, x)

. For all

p_{1}, p_{2}, \dots, p_{d} \in N_{0}

,

E \prod_{i = 1}^{d} {(ξ_{i} - E [ξ_{i}])}^{p_{i}}] = \sum_{ℓ_{1} = 0}^{p_{1}} \dots \sum_{ℓ_{d} = 0}^{p_{d}} \sum_{k_{1} = 0}^{ℓ_{1}} \dots \sum_{k_{d} = 0}^{ℓ_{d}} m^{(\sum_{i = 1}^{d} k_{i})} {(- m)}^{\sum_{i = 1}^{d} (p_{i} - ℓ_{i})} \prod_{i = 1}^{d} \binom{p_{i}}{ℓ_{i}}) \binom{ℓ_{i}}{k_{i}}\} x_{i}^{p_{i} - ℓ_{i} + k_{i}},

(9)

where

(\binom{p}{ℓ})

denotes the binomial coefficient

\frac{p!}{ℓ! (p - ℓ)!}

.

Proof.

By applying the binomial formula to each factor

{(ξ_{i} - E [ξ_{i}])}^{p_{i}}

and using the fact that

E [ξ_{i}] = m x_{i}

for all

i \in {1, 2, \dots, d}

, note that

E \prod_{i = 1}^{d} {(ξ_{i} - E [ξ_{i}])}^{p_{i}}] = \sum_{ℓ_{1} = 0}^{p_{1}} \dots \sum_{ℓ_{d} = 0}^{p_{d}} E \prod_{i = 1}^{d} ξ_{i}^{ℓ_{i}}] \cdot \prod_{i = 1}^{d} \binom{p_{i}}{ℓ_{i}}) {(- m x_{i})}^{p_{i} - ℓ_{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{matrix} \binom{0}{0}\} = 1, \\ \binom{1}{0}\} = 0, \binom{1}{1}\} = 1, \\ \binom{2}{0}\} = 0, \binom{2}{1}\} = 1, \binom{2}{2}\} = 1, \\ \binom{3}{0}\} = 0, \binom{3}{1}\} = 1, \binom{3}{2}\} = 3, \binom{3}{3}\} = 1, \\ \binom{4}{0}\} = 0, \binom{4}{1}\} = 1, \binom{4}{2}\} = 7, \binom{4}{3}\} = 6, \binom{4}{4}\} = 1, \\ \binom{5}{0}\} = 0, \binom{5}{1}\} = 1, \binom{5}{2}\} = 15, \binom{5}{3}\} = 25, \binom{5}{4}\} = 10, \binom{5}{5}\} = 1, \\ \binom{6}{0}\} = 0, \binom{6}{1}\} = 1, \binom{6}{2}\} = 31, \binom{6}{3}\} = 90, \binom{6}{4}\} = 65, \binom{6}{5}\} = 15, \binom{6}{6}\} = 1, \\ \binom{7}{0}\} = 0, \binom{7}{1}\} = 1, \binom{7}{2}\} = 63, \binom{7}{3}\} = 301, \binom{7}{4}\} = 350, \binom{7}{5}\} = 140, \binom{7}{6}\} = 21, \binom{7}{7}\} = 1, \\ \binom{8}{0}\} = 0, \binom{8}{1}\} = 1, \binom{8}{2}\} = 127, \binom{8}{3}\} = 966, \binom{8}{4}\} = 1701, \binom{8}{5}\} = 1050, \binom{8}{6}\} = 266, \binom{8}{7}\} = 28, \binom{8}{8}\} = 1 . \end{matrix}

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

\{\binom{p_{i}}{k_{i}}\}

that are equal to 0, we get the following results directly.

Order 1: For any

j_{1} \in {1, 2, \dots, d}

,

\begin{matrix} E [ξ_{j_{1}}] & = x_{j_{1}} m . \end{matrix}

(11)

Order 2: For any distinct

j_{1}, j_{2} \in {1, 2, \dots, d}

,

\begin{matrix} E [ξ_{j_{1}}^{2}] & = x_{j_{1}} [m + m^{(2)} x_{j_{1}}], \end{matrix}

(12)

\begin{matrix} E [ξ_{j_{1}} ξ_{j_{2}}] & = x_{j_{1}} x_{j_{2}} m^{(2)} . \end{matrix}

(13)

Order 3: For any distinct

j_{1}, j_{2}, j_{3} \in {1, 2, \dots, d}

,

\begin{matrix} E [ξ_{j_{1}}^{3}] & = x_{j_{1}} [m + 3 m^{(2)} x_{j_{1}} + m^{(3)} x_{j_{1}}^{2}], \end{matrix}

(14)

\begin{matrix} E [ξ_{j_{1}}^{2} ξ_{j_{2}}] & = x_{j_{1}} x_{j_{2}} [m^{(2)} + m^{(3)} x_{j_{1}}], \end{matrix}

(15)

\begin{matrix} E [ξ_{j_{1}} ξ_{j_{2}} ξ_{j_{3}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} m^{(3)} . \end{matrix}

(16)

Order 4: For any distinct

j_{1}, j_{2}, j_{3}, j_{4} \in {1, 2, \dots, d}

,

\begin{matrix} E [ξ_{j_{1}}^{4}] & = x_{j_{1}} [m + 7 m^{(2)} x_{j_{1}} + 6 m^{(3)} x_{j_{1}}^{2} + m^{(4)} x_{j_{1}}^{3}], \end{matrix}

(17)

\begin{matrix} E [ξ_{j_{1}}^{3} ξ_{j_{2}}] & = x_{j_{1}} x_{j_{2}} [m^{(2)} + 3 m^{(3)} x_{j_{1}} + m^{(4)} x_{j_{1}}^{2}], \end{matrix}

(18)

\begin{matrix} E [ξ_{j_{1}}^{2} ξ_{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}}], \end{matrix}

(19)

\begin{matrix} E [ξ_{j_{1}}^{2} ξ_{j_{2}} ξ_{j_{3}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}}] m^{(3)} + m^{(4)} x_{j_{1}}], \end{matrix}

(20)

\begin{matrix} E [ξ_{j_{1}} ξ_{j_{2}} ξ_{j_{3}} ξ_{j_{4}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} x_{j_{4}} m^{(4)} . \end{matrix}

(21)

Order 5: For any distinct

j_{1}, j_{2}, j_{3}, j_{4}, j_{5} \in {1, 2, \dots, d}

,

\begin{matrix} E [ξ_{j_{1}}^{5}] & = x_{j_{1}} [m + 15 m^{(2)} x_{j_{1}} + 25 m^{(3)} x_{j_{1}}^{2} + 10 m^{(4)} x_{j_{1}}^{3} + m^{(5)} x_{j_{1}}^{4}], \end{matrix}

(22)

\begin{matrix} E [ξ_{j_{1}}^{4} ξ_{j_{2}}] & = x_{j_{1}} x_{j_{2}} [m^{(2)} + 7 m^{(3)} x_{j_{1}} + 6 m^{(4)} x_{j_{1}}^{2} + m^{(5)} x_{j_{1}}^{3}], \end{matrix}

(23)

\begin{matrix} E [ξ_{j_{1}}^{3} ξ_{j_{2}}^{2}] & = x_{j_{1}} x_{j_{2}} [m^{(2)} + m^{(3)} (3 x_{j_{1}} + x_{j_{2}}) + m^{(4)} (x_{j_{1}}^{2} + 3 x_{j_{1}} x_{j_{2}}) + m^{(5)} x_{j_{1}}^{2} x_{j_{2}}], \end{matrix}

(24)

\begin{matrix} E [ξ_{j_{1}}^{3} ξ_{j_{2}} ξ_{j_{3}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} [m^{(3)} + 3 m^{(4)} x_{j_{1}} + m^{(5)} x_{j_{1}}^{2}], \end{matrix}

(25)

\begin{matrix} E [ξ_{j_{1}}^{2} ξ_{j_{2}}^{2} ξ_{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}}], \end{matrix}

(26)

\begin{matrix} E [ξ_{j_{1}}^{2} ξ_{j_{2}} ξ_{j_{3}} ξ_{j_{4}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} x_{j_{4}} [m^{(4)} + m^{(5)} x_{j_{1}}], \end{matrix}

(27)

\begin{matrix} E [ξ_{j_{1}} ξ_{j_{2}} ξ_{j_{3}} ξ_{j_{4}} ξ_{j_{5}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} x_{j_{4}} x_{j_{5}} m^{(5)} . \end{matrix}

(28)

Order 6: For any distinct

j_{1}, j_{2}, j_{3}, j_{4}, j_{5}, j_{6} \in {1, 2, \dots, d}

,

\begin{matrix} E [ξ_{j_{1}}^{6}] & = x_{j_{1}} [m + 31 m^{(2)} x_{j_{1}} + 90 m^{(3)} x_{j_{1}}^{2} + 65 m^{(4)} x_{j_{1}}^{3} + 15 m^{(5)} x_{j_{1}}^{4} + m^{(6)} x_{j_{1}}^{5}], \end{matrix}

(29)

\begin{matrix} E [ξ_{j_{1}}^{5} ξ_{j_{2}}] & = x_{j_{1}} x_{j_{2}} [m^{(2)} + 15 m^{(3)} x_{j_{1}} + 25 m^{(4)} x_{j_{1}}^{2} + 10 m^{(5)} x_{j_{1}}^{3} + m^{(6)} x_{j_{1}}^{4}], \end{matrix}

(30)

\begin{matrix} E [ξ_{j_{1}}^{4} ξ_{j_{2}}^{2}] & = x_{j_{1}} x_{j_{2}} \begin{matrix} m^{(2)} + m^{(3)} (7 x_{j_{1}} + x_{j_{2}}) + m^{(4)} (6 x_{j_{1}}^{2} + 7 x_{j_{1}} x_{j_{2}}) \\ + m^{(5)} (x_{j_{1}}^{3} + 6 x_{j_{1}}^{2} x_{j_{2}}) + m^{(6)} x_{j_{1}}^{3} x_{j_{2}} \end{matrix}], \end{matrix}

(31)

\begin{matrix} E [ξ_{j_{1}}^{4} ξ_{j_{2}} ξ_{j_{3}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} [m^{(3)} + 7 m^{(4)} x_{j_{1}} + 6 m^{(5)} x_{j_{1}}^{2} + m^{(6)} x_{j_{1}}^{3}], \end{matrix}

(32)

\begin{matrix} E [ξ_{j_{1}}^{3} ξ_{j_{2}}^{3}] & = x_{j_{1}} x_{j_{2}} \begin{matrix} m^{(2)} + m^{(3)} (3 x_{j_{1}} + 3 x_{j_{2}}) + m^{(4)} (x_{j_{1}}^{2} + 9 x_{j_{1}} x_{j_{2}} + x_{j_{2}}^{2}) \\ + m^{(5)} (3 x_{j_{1}}^{2} x_{j_{2}} + 3 x_{j_{1}} x_{j_{2}}^{2}) + m^{(6)} x_{j_{1}}^{2} x_{j_{2}}^{2} \end{matrix}], \end{matrix}

(33)

\begin{matrix} E [ξ_{j_{1}}^{3} ξ_{j_{2}}^{2} ξ_{j_{3}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} [m^{(3)} + m^{(4)} (3 x_{j_{1}} + x_{j_{2}}) + m^{(5)} (x_{j_{1}}^{2} + 3 x_{j_{1}} x_{j_{2}}) + m^{(6)} x_{j_{1}}^{2} x_{j_{2}}], \end{matrix}

(34)

\begin{matrix} E [ξ_{j_{1}}^{3} ξ_{j_{2}} ξ_{j_{3}} ξ_{j_{4}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} x_{j_{4}} [m^{(4)} + 3 m^{(5)} x_{j_{1}} + m^{(6)} x_{j_{1}}^{2}], \end{matrix}

(35)

\begin{matrix} E [ξ_{j_{1}}^{2} ξ_{j_{2}}^{2} ξ_{j_{3}}^{2}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} \begin{matrix} m^{(3)} + m^{(4)} (x_{j_{1}} + x_{j_{2}} + x_{j_{3}}) \\ + 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}} \end{matrix}], \end{matrix}

(36)

\begin{matrix} E [ξ_{j_{1}}^{2} ξ_{j_{2}}^{2} ξ_{j_{3}} ξ_{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}}], \end{matrix}

(37)

\begin{matrix} E [ξ_{j_{1}}^{2} ξ_{j_{2}} ξ_{j_{3}} ξ_{j_{4}} ξ_{j_{5}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} x_{j_{4}} x_{j_{5}} [m^{(5)} + m^{(6)} x_{j_{1}}], \end{matrix}

(38)

\begin{matrix} E [ξ_{j_{1}} ξ_{j_{2}} ξ_{j_{3}} ξ_{j_{4}} ξ_{j_{5}} ξ_{j_{6}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} x_{j_{4}} x_{j_{5}} x_{j_{6}} m^{(6)} . \end{matrix}

(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{matrix} E [ξ_{j_{1}}^{7}] & = x_{j_{1}} \begin{matrix} m + 63 m^{(2)} x_{j_{1}} + 301 m^{(3)} x_{j_{1}}^{2} + 350 m^{(4)} x_{j_{1}}^{3} \\ + 140 m^{(5)} x_{j_{1}}^{4} + 21 m^{(6)} x_{j_{1}}^{5} + m^{(7)} x_{j_{1}}^{6} \end{matrix}], \end{matrix}

(40)

\begin{matrix} E [ξ_{j_{1}}^{6} ξ_{j_{2}}] & = x_{j_{1}} x_{j_{2}} \begin{matrix} m^{(2)} + 31 m^{(3)} x_{j_{1}} + 90 m^{(4)} x_{j_{1}}^{2} \\ + 65 m^{(5)} x_{j_{1}}^{3} + 15 m^{(6)} x_{j_{1}}^{4} + m^{(7)} x_{j_{1}}^{5} \end{matrix}], \end{matrix}

(41)

\begin{matrix} E [ξ_{j_{1}}^{5} ξ_{j_{2}}^{2}] & = x_{j_{1}} x_{j_{2}} \begin{matrix} m^{(2)} + m^{(3)} (15 x_{j_{1}} + x_{j_{2}}) + m^{(4)} (25 x_{j_{1}}^{2} + 15 x_{j_{1}} x_{j_{2}}) \\ + m^{(5)} (10 x_{j_{1}}^{3} + 25 x_{j_{1}}^{2} x_{j_{2}}) + m^{(6)} (x_{j_{1}}^{4} + 10 x_{j_{1}}^{3} x_{j_{2}}) + m^{(7)} x_{j_{1}}^{4} x_{j_{2}} \end{matrix}], \end{matrix}

(42)

\begin{matrix} E [ξ_{j_{1}}^{5} ξ_{j_{2}} ξ_{j_{3}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} [m^{(3)} + 15 m^{(4)} x_{j_{1}} + 25 m^{(5)} x_{j_{1}}^{2} + 10 m^{(6)} x_{j_{1}}^{3} + m^{(7)} x_{j_{1}}^{4}], \end{matrix}

(43)

\begin{matrix} E [ξ_{j_{1}}^{4} ξ_{j_{2}}^{3}] & = x_{j_{1}} x_{j_{2}} \begin{matrix} m^{(2)} + m^{(3)} (7 x_{j_{1}} + 3 x_{j_{2}}) + m^{(4)} (6 x_{j_{1}}^{2} + 21 x_{j_{1}} x_{j_{2}} + x_{j_{2}}^{2}) \\ + m^{(5)} (x_{j_{1}}^{3} + 18 x_{j_{1}}^{2} x_{j_{2}} + 7 x_{j_{1}} x_{j_{2}}^{2}) \\ + m^{(6)} (3 x_{j_{1}}^{3} x_{j_{2}} + 6 x_{j_{1}}^{2} x_{j_{2}}^{2}) + m^{(7)} x_{j_{1}}^{3} x_{j_{2}}^{2} \end{matrix}], \end{matrix}

(44)

\begin{matrix} E [ξ_{j_{1}}^{4} ξ_{j_{2}}^{2} ξ_{j_{3}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} \begin{matrix} m^{(3)} + m^{(4)} (7 x_{j_{1}} + x_{j_{2}}) + m^{(5)} (6 x_{j_{1}}^{2} + 7 x_{j_{1}} x_{j_{2}}) \\ + m^{(6)} (x_{j_{1}}^{3} + 6 x_{j_{1}}^{2} x_{j_{2}}) + m^{(7)} x_{j_{1}}^{3} x_{j_{2}} \end{matrix}], \end{matrix}

(45)

\begin{matrix} E [ξ_{j_{1}}^{4} ξ_{j_{2}} ξ_{j_{3}} ξ_{j_{4}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} x_{j_{4}} [m^{(4)} + 7 m^{(5)} x_{j_{1}} + 6 m^{(6)} x_{j_{1}}^{2} + m^{(7)} x_{j_{1}}^{3}], \end{matrix}

(46)

\begin{matrix} E [ξ_{j_{1}}^{3} ξ_{j_{2}}^{3} ξ_{j_{3}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} \begin{matrix} m^{(3)} + m^{(4)} (3 x_{j_{1}} + 3 x_{j_{2}}) + m^{(5)} (x_{j_{1}}^{2} + 9 x_{j_{1}} x_{j_{2}} + x_{j_{2}}^{2}) \\ + m^{(6)} (3 x_{j_{1}}^{2} x_{j_{2}} + 3 x_{j_{1}} x_{j_{2}}^{2}) + m^{(7)} x_{j_{1}}^{2} x_{j_{2}}^{2} \end{matrix}], \end{matrix}

(47)

\begin{matrix} E [ξ_{j_{1}}^{3} ξ_{j_{2}}^{2} ξ_{j_{3}}^{2}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} \begin{matrix} m^{(3)} + m^{(4)} (3 x_{j_{1}} + x_{j_{2}} + x_{j_{3}}) \\ + m^{(5)} (x_{j_{1}}^{2} + 3 x_{j_{1}} x_{j_{2}} + 3 x_{j_{1}} x_{j_{3}} + x_{j_{2}} x_{j_{3}}) \\ + m^{(6)} (x_{j_{1}}^{2} x_{j_{2}} + x_{j_{1}}^{2} x_{j_{3}} + 3 x_{j_{1}} x_{j_{2}} x_{j_{3}}) + m^{(7)} x_{j_{1}}^{2} x_{j_{2}} x_{j_{3}} \end{matrix}], \end{matrix}

(48)

\begin{matrix} E [ξ_{j_{1}}^{3} ξ_{j_{2}}^{2} ξ_{j_{3}} ξ_{j_{4}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} x_{j_{4}} \begin{matrix} m^{(4)} + m^{(5)} (3 x_{j_{1}} + x_{j_{2}}) \\ + m^{(6)} (x_{j_{1}}^{2} + 3 x_{j_{1}} x_{j_{2}}) + m^{(7)} x_{j_{1}}^{2} x_{j_{2}} \end{matrix}], \end{matrix}

(49)

\begin{matrix} E [ξ_{j_{1}}^{3} ξ_{j_{2}} ξ_{j_{3}} ξ_{j_{4}} ξ_{j_{5}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} x_{j_{4}} x_{j_{5}} [m^{(5)} + 3 m^{(6)} x_{j_{1}} + m^{(7)} x_{j_{1}}^{2}], \end{matrix}

(50)

\begin{matrix} E [ξ_{j_{1}}^{2} ξ_{j_{2}}^{2} ξ_{j_{3}}^{2} ξ_{j_{4}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} x_{j_{4}} \begin{matrix} m^{(4)} + m^{(5)} (x_{j_{1}} + x_{j_{2}} + x_{j_{3}}) \\ + 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}} \end{matrix}], \end{matrix}

(51)

\begin{matrix} E [ξ_{j_{1}}^{2} ξ_{j_{2}}^{2} ξ_{j_{3}} ξ_{j_{4}} ξ_{j_{5}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} x_{j_{4}} x_{j_{5}} m^{(5)} + m^{(6)} (x_{j_{1}} + x_{j_{2}}) + m^{(7)} x_{j_{1}} x_{j_{2}}], \end{matrix}

(52)

\begin{matrix} E [ξ_{j_{1}}^{2} ξ_{j_{2}} ξ_{j_{3}} ξ_{j_{4}} ξ_{j_{5}} ξ_{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}}], \end{matrix}

(53)

\begin{matrix} E [ξ_{j_{1}} ξ_{j_{2}} ξ_{j_{3}} ξ_{j_{4}} ξ_{j_{5}} ξ_{j_{6}} ξ_{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)} . \end{matrix}

(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{matrix} E [ξ_{j_{1}}^{8}] & = x_{j_{1}} \begin{matrix} m + 127 m^{(2)} x_{j_{1}} + 966 m^{(3)} x_{j_{2}}^{2} + 1701 m^{(4)} x_{j_{1}}^{3} \\ + 1050 m^{(5)} x_{j_{1}}^{4} + 266 m^{(6)} x_{j_{1}}^{5} + 28 m^{(7)} x_{j_{1}}^{6} + m^{(8)} x_{j_{1}}^{7} \end{matrix}], \end{matrix}

(55)

\begin{matrix} E [ξ_{j_{1}}^{7} ξ_{j_{2}}] & = x_{j_{1}} x_{j_{2}} \begin{matrix} m + 63 m^{(3)} x_{j_{1}} + 301 m^{(4)} x_{j_{1}}^{2} + 350 m^{(5)} x_{j_{1}}^{3} \\ + 140 m^{(6)} x_{j_{1}}^{4} + 21 m^{(7)} x_{j_{1}}^{5} + m^{(8)} x_{j_{1}}^{6} \end{matrix}], \end{matrix}

(56)

\begin{matrix} E [ξ_{j_{1}}^{6} ξ_{j_{2}}^{2}] & = x_{j_{1}} x_{j_{2}} \begin{matrix} m^{(2)} + m^{(3)} (31 x_{j_{1}} + x_{j_{2}}) + m^{(4)} (90 x_{j_{1}}^{2} + 31 x_{j_{1}} x_{j_{2}}) \\ + m^{(5)} (65 x_{j_{1}}^{3} + 90 x_{j_{1}}^{2} x_{j_{2}}) + m^{(6)} (15 x_{j_{1}}^{4} + 65 x_{j_{1}}^{3} x_{j_{2}}) \\ + m^{(7)} (x_{j_{1}}^{5} + 15 x_{j_{1}}^{4} x_{j_{2}}) + m^{(8)} x_{j_{1}}^{5} x_{j_{2}} \end{matrix}], \end{matrix}

(57)

\begin{matrix} E [ξ_{j_{1}}^{6} ξ_{j_{2}} ξ_{j_{3}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} \begin{matrix} m^{(3)} + 31 m^{(4)} x_{j_{1}} + 90 m^{(5)} x_{j_{1}}^{2} \\ + 65 m^{(6)} x_{j_{1}}^{3} + 15 m^{(7)} x_{j_{1}}^{4} + m^{(8)} x_{j_{1}}^{5} \end{matrix}], \end{matrix}

(58)

\begin{matrix} E [ξ_{j_{1}}^{5} ξ_{j_{2}}^{3}] & = x_{j_{1}} x_{j_{2}} \begin{matrix} m^{(2)} + m^{(3)} (15 x_{j_{1}} + 3 x_{j_{2}}) + m^{(4)} (25 x_{j_{1}}^{2} + 45 x_{j_{1}} x_{j_{2}} + x_{j_{2}}^{2}) \\ + m^{(5)} (10 x_{j_{1}}^{3} + 75 x_{j_{1}}^{2} x_{j_{2}} + 15 x_{j_{1}} x_{j_{2}}^{2}) \\ + m^{(6)} (x_{j_{4}}^{4} + 30 x_{j_{1}}^{3} x_{j_{2}} + 25 x_{j_{1}}^{2} x_{j_{2}}^{2}) \\ + m^{(7)} (3 x_{j_{1}}^{4} x_{j_{2}} + 10 x_{j_{1}}^{3} x_{j_{2}}^{2}) + m^{(8)} x_{j_{1}}^{4} x_{j_{2}}^{2} \end{matrix}], \end{matrix}

(59)

\begin{matrix} E [ξ_{j_{1}}^{5} ξ_{j_{2}}^{2} ξ_{j_{3}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} \begin{matrix} m^{(3)} + m^{(4)} (15 x_{j_{1}} + x_{j_{2}}) + m^{(5)} (25 x_{j_{1}}^{2} + 15 x_{j_{1}} x_{j_{2}}) \\ + m^{(6)} (10 x_{j_{1}}^{3} + 25 x_{j_{1}}^{2} x_{j_{2}}) + m^{(7)} (x_{j_{1}}^{4} + 10 x_{j_{1}}^{3} x_{j_{2}}) + m^{(8)} x_{j_{1}}^{4} x_{j_{2}} \end{matrix}], \end{matrix}

(60)

\begin{matrix} E [ξ_{j_{1}}^{5} ξ_{j_{2}} ξ_{j_{3}} ξ_{j_{4}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} x_{j_{4}} [m^{(4)} + 15 m^{(5)} x_{j_{1}} + 25 m^{(6)} x_{j_{1}}^{2} + 10 m^{(7)} x_{j_{1}}^{3} + m^{(8)} x_{j_{1}}^{4}], \end{matrix}

(61)

\begin{matrix} E [ξ_{j_{1}}^{4} ξ_{j_{2}}^{4}] & = x_{j_{1}} x_{j_{2}} \begin{matrix} m^{(2)} + m^{(3)} (7 x_{j_{1}} + 7 x_{j_{2}}) + m^{(4)} (6 x_{j_{1}}^{2} + 49 x_{j_{1}} x_{j_{2}} + 6 x_{j_{2}}^{2}) \\ + m^{(5)} (x_{j_{1}}^{3} + 42 x_{j_{1}}^{2} x_{j_{2}} + 42 x_{j_{1}} x_{j_{2}}^{2} + x_{j_{2}}^{3}) \\ + m^{(6)} (7 x_{j_{1}}^{3} x_{j_{2}} + 36 x_{j_{1}}^{2} x_{j_{2}}^{2} + 7 x_{j_{1}} x_{j_{2}}^{3}) \\ + m^{(7)} (6 x_{j_{1}}^{3} x_{j_{2}}^{2} + 6 x_{j_{1}}^{2} x_{j_{2}}^{3}) + m^{(8)} x_{j_{1}}^{3} x_{j_{2}}^{3} \end{matrix}], \end{matrix}

(62)

\begin{matrix} E [ξ_{j_{1}}^{4} ξ_{j_{2}}^{3} ξ_{j_{3}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} \begin{matrix} m^{(3)} + m^{(4)} (7 x_{j_{1}} + 3 x_{j_{2}}) + m^{(5)} (6 x_{j_{1}}^{2} + 21 x_{j_{1}} x_{j_{2}} + x_{j_{2}}^{2}) \\ + m^{(6)} (x_{j_{1}}^{3} + 18 x_{j_{1}}^{2} x_{j_{2}} + 7 x_{j_{1}} x_{j_{2}}^{2}) \\ + m^{(7)} (3 x_{j_{1}}^{3} x_{j_{2}} + 6 x_{j_{1}}^{2} x_{j_{2}}^{2}) + m^{(8)} x_{j_{1}}^{3} x_{j_{2}}^{2} \end{matrix}], \end{matrix}

(63)

\begin{matrix} E [ξ_{j_{1}}^{4} ξ_{j_{2}}^{2} ξ_{j_{3}}^{2}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} \begin{matrix} m^{(3)} + m^{(4)} (7 x_{j_{1}} + x_{j_{2}} + x_{j_{3}}) \\ + m^{(5)} (6 x_{j_{1}}^{2} + 7 x_{j_{1}} x_{j_{2}} + 7 x_{j_{1}} x_{j_{3}} + x_{j_{2}} x_{j_{3}}) \\ + m^{(6)} (x_{j_{1}}^{3} + 6 x_{j_{1}}^{2} x_{j_{2}} + 6 x_{j_{1}}^{2} x_{j_{3}} + 7 x_{j_{1}} x_{j_{2}} x_{j_{3}}) \\ + m^{(7)} (x_{j_{1}}^{3} x_{j_{2}} + x_{j_{1}}^{3} x_{j_{3}} + 6 x_{j_{1}}^{2} x_{j_{2}} x_{j_{3}}) + m^{(8)} x_{j_{1}}^{3} x_{j_{2}} x_{j_{3}} \end{matrix}], \end{matrix}

(64)

\begin{matrix} E [ξ_{j_{1}}^{4} ξ_{j_{2}}^{2} ξ_{j_{3}} ξ_{j_{4}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} x_{j_{4}} \begin{matrix} m^{(4)} + m^{(5)} (7 x_{j_{1}} + x_{j_{2}}) + m^{(6)} (6 x_{j_{1}}^{2} + 7 x_{j_{1}} x_{j_{2}}) \\ + m^{(7)} (x_{j_{1}}^{3} + 6 x_{j_{1}}^{2} x_{j_{2}}) + m^{(8)} x_{j_{1}}^{3} x_{j_{2}} \end{matrix}], \end{matrix}

(65)

\begin{matrix} E [ξ_{j_{1}}^{4} ξ_{j_{2}} ξ_{j_{3}} ξ_{j_{4}} ξ_{j_{5}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} x_{j_{4}} x_{j_{5}} [m^{(5)} + 7 m^{(6)} x_{j_{1}} + 6 m^{(7)} x_{j_{1}}^{2} + m^{(8)} x_{j_{1}}^{3}], \end{matrix}

(66)

\begin{matrix} E [ξ_{j_{1}}^{3} ξ_{j_{2}}^{3} ξ_{j_{3}}^{2}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} \begin{matrix} m^{(3)} + m^{(4)} (3 x_{j_{1}} + 3 x_{j_{2}} + x_{j_{3}}) \\ + m^{(5)} (x_{j_{1}}^{2} + x_{j_{2}}^{2} + 3 x_{j_{1}} x_{j_{3}} + 3 x_{j_{2}} x_{j_{3}} + 9 x_{j_{1}} x_{j_{2}}) \\ + m^{(6)} (x_{j_{1}}^{2} x_{j_{3}} + x_{j_{2}}^{2} x_{j_{3}} + 3 x_{j_{1}}^{2} x_{j_{2}} + 3 x_{j_{1}} x_{j_{2}}^{2} + 9 x_{j_{1}} x_{j_{2}} x_{j_{3}}) \\ + m^{(7)} (x_{j_{1}}^{2} x_{j_{2}}^{2} + 3 x_{j_{1}}^{2} x_{j_{2}} x_{j_{3}} + 3 x_{j_{1}} x_{j_{2}}^{2} x_{j_{3}}) + m^{(8)} x_{j_{1}}^{2} x_{j_{2}}^{2} x_{j_{3}} \end{matrix}], \end{matrix}

(67)

\begin{matrix} E [ξ_{j_{1}}^{3} ξ_{j_{2}}^{3} ξ_{j_{3}} ξ_{j_{4}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} x_{j_{4}} \begin{matrix} m^{(4)} + m^{(5)} (3 x_{j_{1}} + 3 x_{j_{2}}) + m^{(6)} (x_{j_{1}}^{2} + 9 x_{j_{1}} x_{j_{2}} + x_{j_{2}}^{2}) \\ + m^{(7)} (3 x_{j_{1}}^{2} x_{j_{2}} + 3 x_{j_{1}} x_{j_{2}}^{2}) + m^{(8)} x_{j_{1}}^{2} x_{j_{2}}^{2} \end{matrix}], \end{matrix}

(68)

\begin{matrix} E [ξ_{j_{1}}^{3} ξ_{j_{2}}^{2} ξ_{j_{3}}^{2} ξ_{j_{4}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} x_{j_{4}} \begin{matrix} m^{(4)} + m^{(5)} (3 x_{j_{1}} + x_{j_{2}} + x_{j_{3}}) \\ + m^{(6)} (3 x_{j_{1}} x_{j_{2}} + 3 x_{j_{1}} x_{j_{3}} + x_{j_{2}} x_{j_{3}}) \\ + m^{(7)} (x_{j_{1}}^{2} x_{j_{2}} + x_{j_{1}}^{2} x_{j_{3}} + 3 x_{j_{1}} x_{j_{2}} x_{j_{3}}) + m^{(8)} x_{j_{1}}^{2} x_{j_{2}} x_{j_{3}} \end{matrix}], \end{matrix}

(69)

\begin{matrix} E [ξ_{j_{1}}^{3} ξ_{j_{2}}^{2} ξ_{j_{3}} ξ_{j_{4}} ξ_{j_{5}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} x_{j_{4}} x_{j_{5}} \begin{matrix} m^{(5)} + m^{(6)} (3 x_{j_{1}} + x_{j_{2}}) \\ + m^{(7)} (x_{j_{1}}^{2} + 3 x_{j_{1}} x_{j_{2}}) + m^{(8)} x_{j_{1}}^{2} x_{j_{2}} \end{matrix}], \end{matrix}

(70)

\begin{matrix} E [ξ_{j_{1}}^{3} ξ_{j_{2}} ξ_{j_{3}} ξ_{j_{4}} ξ_{j_{5}} ξ_{j_{6}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} x_{j_{4}} x_{j_{5}} x_{j_{6}} [m^{(6)} + 3 m^{(7)} x_{j_{1}} + m^{(8)} x_{j_{1}}^{2}], \end{matrix}

(71)

\begin{matrix} E [ξ_{j_{1}}^{2} ξ_{j_{2}}^{2} ξ_{j_{3}}^{2} ξ_{j_{4}}^{2}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} x_{j_{4}} \begin{matrix} m^{(4)} + m^{(5)} (x_{j_{1}} + x_{j_{2}} + x_{j_{3}} + x_{j_{4}}) \\ + 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}}) \\ + 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}}) \\ + m^{(8)} x_{j_{1}} x_{j_{2}} x_{j_{3}} x_{j_{4}} \end{matrix}], \end{matrix}

(72)

\begin{matrix} E [ξ_{j_{1}}^{2} ξ_{j_{2}}^{2} ξ_{j_{3}}^{2} ξ_{j_{4}} ξ_{j_{5}}] & = x_{j_{1}} x_{j_{2}} x_{j_{3}} x_{j_{4}} x_{j_{5}} \begin{matrix} m^{(5)} + m^{(6)} (x_{j_{1}} + x_{j_{2}} + x_{j_{3}}) \\ + 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}} \end{matrix}], \end{matrix}

(73)

\begin{matrix} E [ξ_{j_{1}}^{2} ξ_{j_{2}}^{2} ξ_{j_{3}} ξ_{j_{4}} ξ_{j_{5}} ξ_{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}}], \end{matrix}

(74)

\begin{matrix} E [ξ_{j_{1}}^{2} ξ_{j_{2}} ξ_{j_{3}} ξ_{j_{4}} ξ_{j_{5}} ξ_{j_{6}} ξ_{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}}], \end{matrix}

(75)

\begin{matrix} E [ξ_{j_{1}} ξ_{j_{2}} ξ_{j_{3}} ξ_{j_{4}} ξ_{j_{5}} ξ_{j_{6}} ξ_{j_{7}} ξ_{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)} . \end{matrix}

(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{matrix} E [{(ξ_{j_{1}} - E [ξ_{j_{1}}])}^{2}] & = E [ξ_{j_{1}}^{2}] - {(E [ξ_{j_{1}}])}^{2} = x_{j_{1}} [m + m^{(2)} x_{j_{1}}] - m^{2} x_{j_{1}}^{2} \\ = \begin{matrix} m x_{j_{1}} (1 - x_{j_{1}}) \end{matrix} \end{matrix}

(77)

\begin{matrix} E [(ξ_{j_{1}} - E [ξ_{j_{1}}]) (ξ_{j_{2}} - E [ξ_{j_{2}}])] & = E [ξ_{j_{1}} ξ_{j_{2}}] - E [ξ_{j_{1}}] E [ξ_{j_{2}}] = m^{(2)} x_{j_{1}} x_{j_{2}} - m x_{j_{1}} m x_{j_{2}} \\ = \begin{matrix} - m x_{j_{1}} x_{j_{2}} \end{matrix} . \end{matrix}

(78)

Order 3: For any distinct

j_{1}, j_{2}, j_{3} \in {1, 2, \dots, d}

,

\begin{matrix} E [{(ξ_{j_{1}} - E [ξ_{j_{1}}])}^{3}] & = E [ξ_{j_{1}}^{3}] - 3 E [ξ_{j_{1}}^{2}] E [ξ_{j_{1}}] + 2 {(E [ξ_{j_{1}}])}^{3} \\ = x_{j_{1}} [m + 3 m^{(2)} x_{j_{1}} + m^{(3)} x_{j_{1}}^{2}] - 3 x_{j_{1}} [m + m^{(2)} x_{j_{1}}] m x_{j_{1}} + 2 m^{3} x_{j_{1}}^{3} \\ = \begin{matrix} m x_{j_{1}} (x_{j_{1}} - 1) (2 x_{j_{1}} - 1) \end{matrix} \end{matrix}

(79)

\begin{matrix} E [{(ξ_{j_{1}} - E [ξ_{j_{1}}])}^{2} (ξ_{j_{2}} - E [ξ_{j_{2}}])] \\ = E [ξ_{j_{1}}^{2} ξ_{j_{2}}] - E [ξ_{j_{1}}^{2}] E [ξ_{j_{2}}] - 2 E [ξ_{j_{1}} ξ_{j_{2}}] E [ξ_{j_{1}}] + 2 {(E [ξ_{j_{1}}])}^{2} E [ξ_{j_{2}}] \\ = x_{j_{1}} x_{j_{2}} [m^{(2)} + m^{(3)} x_{j_{1}}] - x_{j_{1}} m + m^{(2)} x_{j_{1}}] m x_{j_{2}} - 2 m^{(2)} x_{j_{1}} x_{j_{2}} m x_{j_{1}} + 2 m^{2} x_{j_{1}}^{2} m x_{j_{2}} \\ = \begin{matrix} m x_{j_{1}} x_{j_{2}} (2 x_{j_{1}} - 1) \end{matrix} \end{matrix}

(80)

\begin{matrix} E [(ξ_{j_{1}} - E [ξ_{j_{1}}]) (ξ_{j_{2}} - E [ξ_{j_{2}}]) (ξ_{j_{3}} - E [ξ_{j_{3}}])] \\ = E [ξ_{j_{1}} ξ_{j_{2}} ξ_{j_{3}}] - E [ξ_{j_{1}} ξ_{j_{2}}] E [ξ_{j_{3}}] - E [ξ_{j_{1}} ξ_{j_{3}}] E [ξ_{j_{2}}] - E [ξ_{j_{2}} ξ_{j_{3}}] E [ξ_{j_{1}}] + 2 E [ξ_{j_{1}}] E [ξ_{j_{2}}] E [ξ_{j_{3}}] \\ = 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}} + 2 m^{3} x_{j_{1}} x_{j_{2}} x_{j_{3}} \\ = \begin{matrix} 2 m x_{j_{1}} x_{j_{2}} x_{j_{3}} \end{matrix} . \end{matrix}

(81)

\begin{matrix} E [{(ξ_{j_{1}} - E [ξ_{j_{1}}])}^{4}] \\ = E [ξ_{j_{1}}^{4}] - 4 E [ξ_{j_{1}}^{3}] E [ξ_{j_{1}}] + 6 E [ξ_{j_{1}}^{2}] {(E [ξ_{j_{1}}])}^{2} - 3 {(E [ξ_{j_{1}}])}^{4} \\ = x_{j_{1}} [m + 7 m^{(2)} x_{j_{1}} + 6 m^{(3)} x_{j_{1}}^{2} + m^{(4)} x_{j_{1}}^{3}] - 4 x_{j_{1}} [m + 3 m^{(2)} x_{j_{1}} + m^{(3)} x_{j_{1}}^{2}] m x_{j_{1}} \\ + 6 x_{j_{1}} [m + m^{(2)} x_{j_{1}}] {(m x_{j_{1}})}^{2} - 3 m^{4} x_{j_{1}}^{4} \\ = \begin{matrix} 3 m^{2} x_{j_{1}}^{2} {(x_{j_{1}} - 1)}^{2} + m x_{j_{1}} (1 - x_{j_{1}}) (6 x_{j_{1}}^{2} - 6 x_{j_{1}} + 1) \end{matrix} \end{matrix}

(82)

Order 4: For any distinct

j_{1}, j_{2}, j_{3}, j_{4} \in {1, 2, \dots, d}

,

\begin{matrix} E [{(ξ_{j_{1}} - E [ξ_{j_{1}}])}^{3} (ξ_{j_{2}} - E [ξ_{j_{2}}])] \\ = E [ξ_{j_{1}}^{3} ξ_{j_{2}}] - E [ξ_{j_{1}}^{3}] E [ξ_{j_{2}}] - 3 E [ξ_{j_{1}}^{2} ξ_{j_{2}}] E [ξ_{j_{1}}] + 3 E [ξ_{j_{1}}^{2}] E [ξ_{j_{1}}] E [ξ_{j_{2}}] \\ + 3 E [ξ_{j_{1}} ξ_{j_{2}}] {(E [ξ_{j_{1}}])}^{2} - 3 {(E [ξ_{j_{1}}])}^{3} E [ξ_{j_{2}}] \\ = x_{j_{1}} x_{j_{2}} [m^{(2)} + 3 m^{(3)} x_{j_{1}} + m^{(4)} x_{j_{1}}^{2}] - x_{j_{1}} [m + 3 m^{(2)} x_{j_{1}} + m^{(3)} x_{j_{1}}^{2}] m x_{j_{2}} \\ - 3 x_{j_{1}} x_{j_{2}} [m^{(2)} + m^{(3)} x_{j_{1}}] m x_{j_{1}} + 3 x_{j_{1}} [m + m^{(2)} x_{j_{1}}] m x_{j_{1}} m x_{j_{2}} + 3 m^{(2)} x_{j_{1}} x_{j_{2}} m^{2} x_{j_{1}}^{2} - 3 m^{3} x_{j_{1}}^{3} m x_{j_{2}} \\ = \begin{matrix} m x_{j_{1}} x_{j_{2}} (3 (m - 2) x_{j_{1}} (x_{j_{1}} - 1) - 1) \end{matrix} \end{matrix}

(83)

\begin{matrix} E [{(ξ_{j_{1}} - E [ξ_{j_{1}}])}^{2} {(ξ_{j_{2}} - E [ξ_{j_{2}}])}^{2}] \\ = E [ξ_{j_{1}}^{2} ξ_{j_{2}}^{2}] - 2 E [ξ_{j_{1}}^{2} ξ_{j_{2}}] E [ξ_{j_{2}}] - 2 E [ξ_{j_{1}} ξ_{j_{2}}^{2}] E [ξ_{j_{1}}] + E [ξ_{j_{1}}^{2}] {(E [ξ_{j_{2}}])}^{2} + E [ξ_{j_{2}}^{2}] {(E [ξ_{j_{1}}])}^{2} \\ + 4 E [ξ_{j_{1}} ξ_{j_{2}}] E [ξ_{j_{1}}] E [ξ_{j_{2}}] - 3 {(E [ξ_{j_{1}}])}^{2} {(E [ξ_{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}}] - 2 x_{j_{1}} x_{j_{2}} [m^{(2)} + m^{(3)} x_{j_{1}}] m x_{j_{2}} \\ - 2 x_{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} \\ + 4 m^{(2)} x_{j_{1}} x_{j_{2}} m x_{j_{1}} m x_{j_{2}} - 3 m^{2} x_{j_{1}}^{2} m^{2} x_{j_{2}}^{2} \\ = \begin{matrix} m (m - 2) x_{j_{1}} x_{j_{2}} (3 x_{j_{1}} x_{j_{2}} - (x_{j_{1}} + x_{j_{2}}) + 1) + m x_{j_{1}} x_{j_{2}} \end{matrix} \end{matrix}

(84)

\begin{matrix} E [{(ξ_{j_{1}} - E [ξ_{j_{1}}])}^{2} (ξ_{j_{2}} - E [ξ_{j_{2}}]) (ξ_{j_{3}} - E [ξ_{j_{3}}])] \\ = E [ξ_{j_{1}}^{2} ξ_{j_{2}} ξ_{j_{3}}] - E [ξ_{j_{1}}^{2} ξ_{j_{2}}] E [ξ_{j_{3}}] - E [ξ_{j_{1}}^{2} ξ_{j_{3}}] E [ξ_{j_{2}}] - 2 E [ξ_{j_{1}} ξ_{j_{2}} ξ_{j_{3}}] E [ξ_{j_{1}}] + E [ξ_{j_{1}}^{2}] E [ξ_{j_{2}}] E [ξ_{j_{3}}] \\ + 2 E [ξ_{j_{1}} ξ_{j_{2}}] E [ξ_{j_{1}}] E [ξ_{j_{3}}] + 2 E [ξ_{j_{1}} ξ_{j_{3}}] E [ξ_{j_{1}}] E [ξ_{j_{2}}] + E [ξ_{j_{2}} ξ_{j_{3}}] {(E [ξ_{j_{1}}])}^{2} - 3 {(E [ξ_{j_{1}}])}^{2} E [ξ_{j_{2}}] E [ξ_{j_{3}}] \\ = 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}} \\ - 2 m^{(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}} + 2 m^{(2)} x_{j_{1}} x_{j_{2}} m x_{j_{1}} m x_{j_{3}} \\ + 2 m^{(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} - 3 m^{2} x_{j_{1}}^{2} m x_{j_{2}} m x_{j_{3}} \\ = \begin{matrix} m (m - 2) x_{j_{1}} x_{j_{2}} x_{j_{3}} (3 x_{j_{1}} - 1) \end{matrix} \end{matrix}

(85)

\begin{matrix} E [(ξ_{j_{1}} - E [ξ_{j_{1}}]) (ξ_{j_{2}} - E [ξ_{j_{2}}]) (ξ_{j_{3}} - E [ξ_{j_{3}}]) (ξ_{j_{4}} - E [ξ_{j_{4}}])] \\ = E [ξ_{j_{1}} ξ_{j_{2}} ξ_{j_{3}} ξ_{j_{4}}] - E [ξ_{j_{1}} ξ_{j_{2}} ξ_{j_{3}}] E [ξ_{j_{4}}] - E [ξ_{j_{1}} ξ_{j_{2}} ξ_{j_{4}}] E [ξ_{j_{3}}] - E [ξ_{j_{1}} ξ_{j_{3}} ξ_{j_{4}}] E [ξ_{j_{2}}] - E [ξ_{j_{2}} ξ_{j_{3}} ξ_{j_{4}}] E [ξ_{j_{1}}] \\ + E [ξ_{j_{1}} ξ_{j_{2}}] E [ξ_{j_{3}}] E [ξ_{j_{4}}] + E [ξ_{j_{1}} ξ_{j_{3}}] E [ξ_{j_{2}}] E [ξ_{j_{4}}] + E [ξ_{j_{1}} ξ_{j_{4}}] E [ξ_{j_{2}}] E [ξ_{j_{3}}] \\ + E [ξ_{j_{2}} ξ_{j_{3}}] E [ξ_{j_{1}}] E [ξ_{j_{4}}] + E [ξ_{j_{2}} ξ_{j_{4}}] E [ξ_{j_{1}}] E [ξ_{j_{3}}] + E [ξ_{j_{3}} ξ_{j_{4}}] E [ξ_{j_{1}}] E [ξ_{j_{2}}] - 3 E [ξ_{j_{1}}] E [ξ_{j_{2}}] E [ξ_{j_{3}}] E [ξ_{j_{4}}] \\ = 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}} \\ + 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}} \\ + 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}} - 3 m^{4} x_{j_{1}} x_{j_{2}} x_{j_{3}} x_{j_{4}} \\ = \begin{matrix} 3 m (m - 2) x_{j_{1}} x_{j_{2}} x_{j_{3}} x_{j_{4}} \end{matrix} . \end{matrix}

(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].

Author Contributions

All contributions were made by the sole author of the article, F.O. The author has read and agreed to the published version of the manuscript.

Funding

This research was funded by a postdoctoral fellowship from the NSERC (PDF) and a postdoctoral fellowship supplement from the FRQNT (B3X).

Acknowledgments

We thank the three referees for their comments.

Conflicts of Interest

The author declares no conflict of interest.

Abbreviations

The following abbreviations are used in this manuscript:

i.i.d.

independent and identically distributed

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

Abstract

1. The Multinomial Distribution

2. Motivation

3. Results

4. Numerical Implementation

5. Explicit Formulas

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

5.2. Computation of the Central Moments Up to Order 4

6. Conclusions

Author Contributions

Funding

Acknowledgments

Conflicts of Interest

Abbreviations

References

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