Exploring the Higher-Order Moment Behavior of Certain Probability Distributions by Combinatorial and Asymptotic Analysis

Abstract

For the binomial, Poisson and normal distributions, the n-th central-raw ratios $\mathbf{E}{(\xi -\mathbf{E}\xi )}^n/\mathbf{E}{\xi }^n$ are known to be infinitesimal with distinct patterns. In this paper, the higher-order moments of the negative binomial distribution $NB(r,p)$ and the Gamma distribution $Ga(\alpha ,\beta )$ are derived using combinatorial and asymptotic methods. Consequently, the asymptotic central-raw ratios of these distributions are shown to be non-zero constants: $q^{{\displaystyle \frac{q}{p}}r}$ (where $q=1-p$) for $NB(r,p)$ and $e^{-\alpha }$ for $Ga(\alpha ,\beta )$ as $n\to +\infty$.

1. Introduction

Moments of random variables have many important applications in probability and statistics. Numerous works focus on the first four moments: mean, variance, skewness and kurtosis. For higher-order moments of a random variable $\xi$, namely the n-th raw moment $\mathbf{E}{\xi }^n$ and the n-th central moment $\mathbf{E}{(\xi -\mathbf{E}\xi )}^n$, fewer results are available in the literature, with the exception of the Beta, Weibull and Gamma distributions, which have simple closed-form expressions for their n-th raw moments [1,2,3].

Some results on the higher-order moments of common probability distributions are found in combinatorics and number theory. The n-th raw moment of the Poisson distribution $\mathcal{P}\left(1\right)$ is known to be the n-th Bell number $B_n$, which counts the number of ways to partition a set of n elements into nonempty subsets [4] (p. 160) [5] (A000110). That is,

$$\begin{array}{c}\hfill \mathbf{E}{\mathcal{P}}^n\left(1\right)=B_n={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{k^n}{k!}}{e}^{-1},n\ge 0.\end{array}$$

Graham, Knuth and Patashnik derived an asymptotic estimate for Bell numbers by bounding the defining infinite sum [6] (p. 493)

$$\begin{array}{c}\hfill B_n\sim {\displaystyle \frac{m_n^n}{\sqrt{\log n}}}{e}^{m_n-n-{\displaystyle \frac{1}{2}}},n\to +\infty ,\end{array}$$

with the scaling parameter $m_n$, solving the fixed-point equation $m_n\log m_n=n-{\displaystyle \frac{1}{2}}$. Separately, Lovász applied central limit arguments for set partitions to produce an alternative large-n approximation, presented as Exercise 9 in Section 1 of [7]:

$$\begin{array}{c}\hfill B_n\sim {\displaystyle \frac{1}{\sqrt{n}}}{\lambda }_n^{n+{\displaystyle \frac{1}{2}}}{e}^{{\lambda }_n-n-1},n\to +\infty ,\end{array}$$

where ${\lambda }_n$ satisfies ${\lambda }_n\log {\lambda }_n=n$. These two distinct approximations characterize the growth of the n-th raw moment of a unit Poisson random variable, whose value coincides with the Bell number $B_n$.

Moreover, the n-th raw moment of the normal distribution $N(1,1)$ is shown to be the number ${\mathcal{T}}_n$ of standard Young tableaux of size n (equivalently, the number of self-inverse permutations on n letters) [8]. H. S. Wilf derived the following asymptotic formula using Hayman’s method [9] (5.41):

$$\begin{array}{c}\hfill {\mathcal{T}}_n\sim {\displaystyle \frac{1}{\sqrt{2}}}{n}^{{\displaystyle \frac{n}{2}}}{e}^{(-{\displaystyle \frac{n}{2}}+\sqrt{n}-{\displaystyle \frac{1}{4}})},n\to +\infty .\end{array}$$

Applying Laplace’s method to the integral, another improved asymptotic formula is given by [8] (Thm.3):

$$\begin{array}{c}\hfill \mathbf{E}{N}^n(1,1)={\mathcal{T}}_n\sim {\displaystyle \frac{{(\frac{1}{2}+\sqrt{n+\frac{1}{4}})}^n}{\sqrt{2}}}{e}^{(-{\displaystyle \frac{n}{2}}+{\displaystyle \frac{\sqrt{n+1}}{2}}-{\displaystyle \frac{1}{4}})},n\to +\infty .\end{array}$$

In this paper we consider the negative binomial distribution $NB(r,p)$ with real parameters $r>0$ and $0<p<1$:

$$\begin{array}{c}\hfill \mathbf{P}(NB(r,p)=i)=\left({\displaystyle \genfrac{}{}{0pt}{}{r+i-1}{i}}\right)p^r{q}^i,i\ge 0,q=1-p.\end{array}$$

(1)

Geometric distribution is $NB(1,p)$ and Pascal distribution is $NB(k,p)$ when k is a positive integer. The moment generating function (m.g.f.) of $NB(r,p)$ is known to be [2] (p. 139)

$$\begin{array}{c}\hfill M\left(t\right)=\mathbf{E}{e}^{tNB(r,p)}={\displaystyle \sum _{n=0}^{\infty }}\mathbf{E}{\left[NB(r,p)\right]}^n{\displaystyle \frac{t^n}{n!}}={\left({\displaystyle \frac{p}{1-q{e}^t}}\right)}^r.\end{array}$$

(2)

Negative binomial distribution is commonly used to describe the distribution of count data [10]. For the special case of geometric distribution $NB(1,{\displaystyle \frac{1}{2}})$, its m.g.f. is known to be ${\displaystyle \frac{1}{2-e^t}}$ from equality (2). However, ${\displaystyle \frac{1}{2-e^t}}$ is also the exponential generating function of the n-th ordered Bell number $b_n$ (also known as a Fubini number [5] (A000670)). Since ${\displaystyle \frac{1}{2-e^t}}$ has the simple pole with smallest modulus $t_0=\log 2$, H. S. Wilf applied the poles method to derive the following highly accurate asymptotic formula [9] (p. 189) (see also (26) in Section 3):

$$\begin{array}{c}\hfill b_n\sim {\displaystyle \frac{n!}{2{\left(\log 2\right)}^{n+1}}},n\to +\infty .\end{array}$$

(3)

Hence, we know that the n-th raw moment of geometric distribution with parameter $p={\displaystyle \frac{1}{2}}$ is the n-th ordered Bell number, and (3) is a nice approximation to $\mathbf{E}{\left[NB(1,{\displaystyle \frac{1}{2}})\right]}^n$.

Notice that the n-th Bell number satisfies $B_n={\sum }_{k=0}^{n}S(n,k)$ and the n-th ordered Bell number satisfies $b_n={\sum }_{k=0}^{n}k!S(n,k)$, where $S(n,k)$ are the Stirling numbers of the second kind. It is known that the raw moments of both the Poisson distribution $\mathcal{P}\left(\lambda \right)$ and the binomial distribution $B(N,p)$ are related to the Stirling numbers of the second kind [4] (p. 160):

$$\begin{array}{ccc}\hfill & & \mathbf{E}{\mathcal{P}}^n\left(\lambda \right)={\displaystyle \sum _{k=0}^n}S(n,k){\lambda }^k,n\ge 0.\hfill \end{array}$$

(4)

$$\begin{array}{ccc}& & \mathbf{E}{\left[B(N,p)\right]}^n={\displaystyle \sum _{k=1}^{min(n,N)}}S(n,k){\left(N\right)}_k{p}^k,n\ge 1,\hfill \end{array}$$

(5)

where ${\left(x\right)}_k=x(x-1)\dots (x-k+1),{\left(x\right)}_0=1$ is the falling factorial. In fact, the n-th raw moment of negative binomial distribution $NB(r,p)$ is also involved in $S(n,k)$ (see (12) below).

We introduce the central-raw ratio statistic $r_n\left(\xi \right)=\mathbf{E}{(\xi -\mathbf{E}\xi )}^n/\mathbf{E}{\xi }^n$ to quantify the relative magnitude of high-order central moments against raw moments. The prior work in [11] established that binomial, Poisson and normal random variables all produce vanishing $r_n$ as the moment order n grows large, while their decay rates differ sharply. Explicit large-n asymptotic forms for these three families are given below:

$$\begin{array}{ccc}& & r_n\left(\mathcal{P}\left(\lambda \right)\right)\sim {\left({\displaystyle \frac{\lambda \log (n/\lambda )}{n}}\right)}^{\lambda },\hfill \end{array}$$

(6)

$$\begin{array}{ccc}& & r_n\left(B(N,p)\right)\sim {(1-p)}^n,0<p<{\displaystyle \frac{1}{2}},\hfill \end{array}$$

(7)

$$\begin{array}{ccc}& & r_n\left(N(\mu ,{\sigma }^2)\right)\sim 2e^{-{\displaystyle \frac{\mu }{\sigma }}\sqrt{n}+{\displaystyle \frac{{\mu }^2}{4{\sigma }^2}}},\mu >0,n\mathrm{even}.\hfill \end{array}$$

(8)

Specializing to unit Poisson variable $\mathcal{P}\left(1\right)$, we simplify the limit expression to obtain

$$\begin{array}{c}\hfill r_n\left(\mathcal{P}\left(1\right)\right)={\displaystyle \frac{\mathbf{E}{[\mathcal{P}\left(1\right)-1]}^n}{\mathbf{E}{\mathcal{P}}^n\left(1\right)}}\sim {\displaystyle \frac{\log n}{n}},n\to +\infty .\end{array}$$

In fact, from the moment generating functions

$$\begin{array}{c}\hfill \mathbf{E}{e}^{t\mathcal{P}\left(1\right)}=e^{(e^t-1)},\mathbf{E}{e}^{t\left(\mathcal{P}\right(1)-1)}=e^{(e^t-1-t)},\end{array}$$

We know that the n-th raw moment $\mathbf{E}{\mathcal{P}}^n\left(1\right)$ of Poisson distribution is the number of ways that a set of n elements can be partitioned into nonempty subsets. In combinatorial enumeration terms, $\mathbf{E}{[\mathcal{P}\left(1\right)-1]}^n$ counts all set partitions of an n-element collection with every block containing two or more entries; this combinatorial sequence is cataloged as OEIS A000296 [5].

Central-raw ratios also have applications in statistics. For a fixed n, the n-th sample central-raw ratio ${\widehat{r}}_{n,m}\left(\xi \right)$ is defined by

$$\begin{array}{c}\hfill {\widehat{r}}_{n,m}\left(\xi \right)={\displaystyle \frac{{\sum }_{i=1}^m{({\xi }_i-\overline{\xi })}^n}{{\sum }_{i=1}^m{\xi }_i^n}}.\end{array}$$

For fixed n, under standard moment conditions, ${\widehat{r}}_{n,m}\left(\xi \right)\to r_n\left(\xi \right)$ almost surely as $m\to \infty$ by the law of large numbers. Hence, if two groups of data have significantly different sample central-raw ratios (e.g., one is infinitesimal while the other tends to a non-zero constant), this may suggest that they come from different populations. A rigorous treatment of such a test is beyond the scope of this paper.

The present results, together with earlier findings for the binomial, Poisson and normal distributions [11], suggest a natural two-category classification of probability distributions according to the limiting behavior of their central-raw ratios: Class I, where the ratio vanishes as $n\to \infty$ (binomial, Poisson, normal), and Class II, where the ratio converges to a positive constant (negative binomial, Gamma). This classification provides a coarse-grained perspective on high-order moment asymptotics and may serve as a starting point for further investigations.

In Section 2 we briefly present the combinatorial and asymptotic tools used in this paper, and give a formula for the n-th raw moment of $NB(r,p)$ similar to (4) and (5). These combinatorial tools include Stirling numbers of the first and second kind, generalized Bernoulli polynomials, and the Lagrange inversion formula. The asymptotic methods include the poles method and Darboux’s method, which are common techniques for approximating generating functions.

In Section 3, the poles method is used to derive asymptotic formulas for the higher-order moments of $NB(r,p)$. The n-th raw moment is approximated by a sum of $\lceil r\rceil$ terms. When r is a positive integer, numerical results demonstrate that the approximation for Pascal distribution is very accurate. The asymptotic central-raw ratio of the negative binomial distribution is shown to be $r_n\left(NB(r,p)\right)\sim q^{{\displaystyle \frac{q}{p}}r}$, which is not infinitesimal. This can help discriminate between samples from a Poisson distribution and those from a negative binomial distribution.

In Section 4 we show that all n-th ($n\ge 2$) central moments of the Gamma $Ga(\alpha ,\beta )$ distribution are positive. Darboux’s method is applied to derive an asymptotic formula for the n-th central moment of the Gamma distribution. The asymptotic central-raw ratio is $r_n\left(Ga(\alpha ,\beta )\right)\sim e^{-\alpha }$, which is not infinitesimal and is independent of $\beta$.

In Section 5 some numerical results are given and Section 6 is the conclusions.

2. Preliminaries

2.1. Stirling Numbers of the First and Second Kind

Stirling numbers of the second kind $S(n,k)$ enumerate all unlabeled k-partitions of an n-element set, where every subset is nonempty. These integers obey a standard two-index recurrence documented in [4] (p. 208):

$$\begin{array}{ccc}\hfill & & S(n,k)=S(n-1,k-1)+kS(n-1,k),n,k\ge 1;\hfill \end{array}$$

(9)

$$\begin{array}{ccc}& & S(n,0)=S(0,k)=0,\mathrm{except}S(0,0)=1,\hfill \end{array}$$

(10)

For fixed non-negative integer k, their exponential generating function takes the compact form below [4] (p. 206):

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{k!}}{(e^t-1)}^k=\sum _{n\ge k}S(n,k){\displaystyle \frac{t^n}{n!}}.\end{array}$$

(11)

Notice that the m.g.f. (2) of negative binomial distribution can be expanded as

$$\begin{array}{ccc}& & {\displaystyle \sum _{n=0}^{\infty }}\mathbf{E}{\left[NB(r,p)\right]}^n{\displaystyle \frac{t^n}{n!}}={\left({\displaystyle \frac{p}{1-q{e}^t}}\right)}^r={\left[1-{\displaystyle \frac{q}{p}}(e^t-1)\right]}^{-r}\hfill \\ & & ={\displaystyle \sum _{k=0}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{-r}{k}}\right){\left(-{\displaystyle \frac{q}{p}}\right)}^k{(e^t-1)}^k={\displaystyle \sum _{k=0}^{\infty }}{\langle r\rangle }_k{\left({\displaystyle \frac{q}{p}}\right)}^k\sum _{n\ge k}S(n,k){\displaystyle \frac{t^n}{n!}}\hfill \\ & & ={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{t^n}{n!}}{\displaystyle \sum _{k=0}^n}S(n,k){\langle r\rangle }_k{\left({\displaystyle \frac{q}{p}}\right)}^k,\hfill \end{array}$$

where ${\langle x\rangle }_k=x(x+1)\dots (x+k-1)$, ${\langle x\rangle }_0=1$ is the rising factorial. Hence, we have the following:

Proposition 1. 

The n-th negative binomial raw moment is

$$\begin{array}{c}\hfill \mathbf{E}{\left[NB(r,p)\right]}^n={\displaystyle \sum _{k=0}^n}S(n,k){\langle r\rangle }_k{\left({\displaystyle \frac{q}{p}}\right)}^k,n\ge 0,\end{array}$$

(12)

and $\mathbf{E}{\left[NB(r,p)\right]}^n={\mu }_n(r,t){|}_{t={\displaystyle \frac{q}{p}}}$, which satisfies the following recurrence:

$$\begin{array}{c}\hfill {\mu }_{n+1}(r,t)=rt{\mu }_n(r+1,t)+t{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}{\mu }_n(r,t),{\mu }_1(r,t)=rt.\end{array}$$

(13)

Recurrence (13) follows from recurrence relation (9) for Stirling numbers of the second kind, since

$$\begin{array}{c}\hfill S(n+1,k){\langle r\rangle }_k{t}^k=rt\left[S(n,k-1){\langle r+1\rangle }_{k-1}{t}^{k-1}\right]+t{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}\left(S(n,k){\langle r\rangle }_k{t}^k\right).\end{array}$$

The Stirling number of the first kind $s(n,k)$ is ${(-1)}^{n-k}$ times the number of permutations of n elements with exactly k cycles, which has the exponential generating function (for fixed $k\ge 0$) [4] (p. 212)

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{k!}}{\log }^k(1+t)=\sum _{n\ge k}s(n,k){\displaystyle \frac{t^n}{n!}}.\end{array}$$

(14)

$\mathfrak{s}(n,k)={(-1)}^{n-k}s(n,k)$ is called the unsigned Stirling number of the first kind, which will be used in the approximation to the raw moments of the negative binomial distribution in Section 3.

Remark 1. 

For the negative binomial distribution, there exist other useful recurrence relations in the literature. Johnson, Kotz and Kemp [10] (p. 216) gave the following recurrence for the central moments $c_n=\mathbf{E}{[NB(r,p)-\mathbf{E}NB(r,p)]}^n$:

$$\begin{array}{c}\hfill c_{n+1}=q{\displaystyle \frac{\mathsf{\partial }{c}_n}{\mathsf{\partial }q}}+{\displaystyle \frac{rnq}{p}}{c}_{n-1},n\ge 1.\end{array}$$

In terms of raw moments $m_n\left(r\right)=\mathbf{E}{\left[NB(r,p)\right]}^n$, differentiating the individual probabilities in (1) with respect to q yields

$$\begin{array}{c}\hfill m_{n+1}\left(r\right)={\displaystyle \frac{rq}{p}}{m}_n\left(r\right)+q{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }q}}{m}_n\left(r\right),\end{array}$$

while differentiating the moment generating function (2) with respect to q gives

$$\begin{array}{c}\hfill p{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }q}}{m}_n\left(r\right)={\displaystyle \frac{r}{q}}\left(m_n(r+1)-m_n\left(r\right)\right).\end{array}$$

Eliminating the derivative term between these identities yields the simpler recursion

$$\begin{array}{c}\hfill m_{n+1}\left(r\right)=r\left[p^{-1}{m}_n(r+1)-m_n\left(r\right)\right].\end{array}$$

These identities are consistent with recurrence (13) obtained from the Stirling numbers of the second kind.

2.2. Lagrange Inversion Formula

We adopt standard generating-function notation: $\left[t^n\right]f\left(t\right)$ stands for the coefficient attached to monomial $t^n$ within the power series expansion of $f\left(t\right)$. If $f\left(t\right)={\sum }_{n\ge 0}{a}_n{t}^n$, we have $\left[t^n\right]f\left(t\right)=a_n$, which immediately yields the identity

$$\begin{array}{c}\hfill \left[{\displaystyle \frac{t^n}{n!}}\right]f\left(t\right)=n!\left[t^n\right]f\left(t\right).\end{array}$$

Lemma 1 

(Lagrange inversion formula [4] (p. 148)). Suppose $f\left(t\right)={\sum }_{n\ge 1}{a}_n{t}^n$ with nonvanishing linear coefficient $a_1\ne 0$. Let $f^{\langle -1\rangle }\left(t\right)$ denote its compositional inverse such that $f\left(f^{-1}\left(t\right)\right)=f^{-1}\left(f\left(t\right)\right)=t$. For any integers satisfying $1\le k\le n$, the coefficient identity holds:

$$\begin{array}{c}\hfill \left[t^n\right]{\left(f^{\langle -1\rangle }\left(t\right)\right)}^k={\displaystyle \frac{k}{n}}\left[t^{n-k}\right]{\left({\displaystyle \frac{f\left(t\right)}{t}}\right)}^{-n}.\end{array}$$

(15)

2.3. Generalized Bernoulli Polynomials

The generalized Bernoulli polynomial $B_n^{\left(\alpha \right)}\left(x\right)$ ($\alpha >0$, $x\in \mathbb{C}$) is defined by the exponential generating function [12] (p. 19)

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{t}{e^t-1}}\right)}^{\alpha }{e}^{xt}={\displaystyle \sum _{n=0}^{\infty }}{B}_n^{\left(\alpha \right)}\left(x\right){\displaystyle \frac{t^n}{n!}}.\end{array}$$

(16)

$B_n^{\left(\alpha \right)}\left(x\right)$ is a polynomial in x of degree n and $B_0^{\left(\alpha \right)}\left(x\right)=1$. $B_n^{\left(\alpha \right)}=B_n^{\left(\alpha \right)}\left(0\right)$ is called the generalized Bernoulli number.

Consider $f\left(t\right)=e^t-1$, whose compositional inverse is $f^{\langle -1\rangle }\left(t\right)=\log (1+t)$. For positive integer k and $0\le n\le k-1$, we apply Lemma 1 with $f\left(t\right)=e^t-1$ and replace k in (15) by $k-n$. This gives

$$\begin{array}{c}\hfill \left[t^k\right]{\left(\log (1+t)\right)}^{k-n}={\displaystyle \frac{k-n}{k}}\left[t^n\right]{\left({\displaystyle \frac{e^t-1}{t}}\right)}^{-k}.\end{array}$$

Consequently,

$$\begin{array}{ccc}\hfill B_n^{\left(k\right)}& =& \left[{\displaystyle \frac{t^n}{n!}}\right]{\left({\displaystyle \frac{t}{e^t-1}}\right)}^k=n!\left[t^n\right]{\left({\displaystyle \frac{e^t-1}{t}}\right)}^{-k}\hfill \\ & =& n!{\displaystyle \frac{k}{k-n}}\left[t^k\right]{\left(\log (1+t)\right)}^{k-n}\hfill \\ & =& n!{\displaystyle \frac{k}{k-n}}{\displaystyle \frac{(k-n)!}{k!}}\left[{\displaystyle \frac{t^k}{k!}}\right]{\displaystyle \frac{1}{(k-n)!}}{\left(\log (1+t)\right)}^{k-n}\hfill \\ & =& {\displaystyle \frac{1}{\left(\genfrac{}{}{0pt}{}{k-1}{n}\right)}}s(k,k-n),\hfill \end{array}$$

where $s(k,k-n)$ is the Stirling number of the first kind, and the last step uses exponential generating function (14). Therefore,

Lemma 2 

(Exercise 18 in [4] (p. 227)). Let k be a positive integer, and the generalized Bernoulli numbers satisfy

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{t}{e^t-1}}\right)}^k={\displaystyle \sum _{n=0}^{\infty }}{B}_n^{\left(k\right)}{\displaystyle \frac{t^n}{n!}}.\end{array}$$

For $0\le n\le k-1$, we have

$$\begin{array}{c}\hfill B_n^{\left(k\right)}={\displaystyle \frac{s(k,k-n)}{\left(\genfrac{}{}{0pt}{}{k-1}{n}\right)}}.\end{array}$$

(17)

2.4. Asymptotic Methods

Let $z_0$ denote the complex point with minimal modulus where $f\left(z\right)$ has a pole of order $k\ge 1$. Inside a punctured open disk centered at $z_0$, the Laurent expansion of $f\left(z\right)$ separates singular regular parts:

$$\begin{array}{c}\hfill f\left(z\right)={\displaystyle \sum _{j=1}^k}{a}_{-j}{(z-z_0)}^{-j}+{\displaystyle \sum _{j=0}^{\infty }}{a}_j{(z-z_0)}^j.\end{array}$$

Lemma 3 

(Poles method, Theorem 5.5 in [9] (p. 189)). Take $z_0$ as the minimal-modulus pole of $f\left(z\right)$ with multiplicity $k\ge 1$, and write R for the modulus of the next closest singularity. For arbitrarily small $\epsilon >0$, the asymptotic coefficient expansion reads

$$\begin{array}{c}\hfill \left[z^n\right]f\left(z\right)={\displaystyle \sum _{j=1}^k}\left({\displaystyle \genfrac{}{}{0pt}{}{n+j-1}{j-1}}\right){\displaystyle \frac{{(-1)}^j{a}_{-j}}{z_0^{n+j}}}+O\left({\left({\displaystyle \frac{1}{R}}+\epsilon \right)}^n\right).\end{array}$$

(18)

Lemma 4 will be used in the approximation to the central moments of the Gamma distribution in Section 4; it is the simple version ($m=1$) of the original theorem (Theorem 5.11 in [9] (p. 194)).

Lemma 4 

(Darboux method). Let $f\left(z\right)$ be analytic in some disk $\left|z\right|<1+\eta$, and suppose that in a neighborhood of $z=1$ it has the expansion $f\left(z\right)={\sum }_{j\ge 0}{a}_j{(1-z)}^j$. Let $\beta \notin \{0,1,2,3,\dots \}$. Then

$$\begin{array}{ccc}\hfill \left[z^n\right]{(1-z)}^{\beta }f\left(z\right)& \sim & \left[z^n\right]\left[a_0{(1-z)}^{\beta }+a_1{(1-z)}^{\beta +1}\right]\hfill \\ & \sim & a_0\left({\displaystyle \genfrac{}{}{0pt}{}{n-\beta -1}{n}}\right)+a_1\left({\displaystyle \genfrac{}{}{0pt}{}{n-\beta -2}{n}}\right),n\to +\infty .\hfill \end{array}$$

(19)

3. Asymptotic Moments for Negative Binomials

When r is a positive integer, Lemma 2 directly gives a nice approximation to the higher-order raw moments of the Pascal distribution. If $r>0$ is not an integer, we need to modify the poles method to obtain a good approximation to the raw moments of the negative binomial distribution.

Let $r_0=\lceil r\rceil -r$, where $\lceil r\rceil =min\{k\in \mathbb{Z}\mid k\ge r\}$.

Theorem 1. 

(1) The n-th raw moment of the negative binomial distribution $NB(r,p)$ has the following asymptotic expansion:

$$\begin{array}{ccc}\hfill & & \mathbf{E}{\left[NB(r,p)\right]}^n\sim {\displaystyle \frac{p^r}{(\lceil r\rceil -1)!}}{\displaystyle \sum _{j=1}^{\lceil r\rceil }}{\displaystyle \frac{(j-1)!{\langle j-r_0\rangle }_n}{{\left(\log (1/q)\right)}^{n+j-r_0}}}\mathfrak{s}(\lceil r\rceil ,j)\hfill \\ & & -{\displaystyle \frac{r_0}{2}}{\displaystyle \frac{p^r}{(\lceil r\rceil -1)!}}{\displaystyle \sum _{j=2}^{\lceil r\rceil }}{\displaystyle \frac{(j-1)!{\langle j-1-r_0\rangle }_n}{{\left(\log (1/q)\right)}^{n+j-1-r_0}}}\mathfrak{s}(\lceil r\rceil ,j),n\to +\infty .\hfill \end{array}$$

(20)

(2) All the n-th ($n\ge 2$) central moments of the negative binomial distribution are positive. The n-th negative binomial central moment has the following asymptotic:

$$\begin{array}{c}\hfill \mathbf{E}[NB(r,p)-{\displaystyle \frac{q}{p}}r{]}^n\sim p^r{q}^{{\displaystyle \frac{q}{p}}r}{\displaystyle \frac{{\langle r\rangle }_n}{{\left(\log (1/q)\right)}^{n+r}}},n\to +\infty .\end{array}$$

(21)

(3) The asymptotic central-raw ratio of negative binomial distribution is

$$\begin{array}{c}\hfill r_n\left(NB(r,p)\right)\sim q^{{\displaystyle \frac{q}{p}}r},n\to +\infty .\end{array}$$

(22)

Proof. 

Intuitively, we first extract the dominant pole term of the NB moment generating function, then expand the singular part near $z_0=\log (1/q)$ and apply the poles method to extract the leading asymptotic coefficients for the raw moments.

(1) Write the m.g.f. of the negative binomial distribution $NB(r,p)$ as

$$\begin{array}{c}\hfill f\left(z\right)=({\displaystyle \frac{p}{1-q{e}^z}}{)}^r={(e^{z-z_0}-1)}^{r_0}\times {\displaystyle \frac{{(-p)}^r}{{(z-z_0)}^{\lceil r\rceil }}}{\left({\displaystyle \frac{z-z_0}{e^{z-z_0}-1}}\right)}^{\lceil r\rceil },\end{array}$$

where $z_0=\log (1/q)$. Since $\lceil r\rceil$ is a positive integer, the pole $z_0=\log (1/q)$ with the smallest modulus of ${\left({\displaystyle \frac{z-z_0}{e^{z-z_0}-1}}\right)}^{\lceil r\rceil }$ has order $\lceil r\rceil$.

First, since $0\le r_0<1$ by definition, we have

$$\begin{array}{c}\hfill {(e^{z-z_0}-1)}^{r_0}\sim {((z-z_0)+{\displaystyle \frac{{(z-z_0)}^2}{2}})}^{r_0}\sim {(z-z_0)}^{r_0}+{\displaystyle \frac{r_0}{2}}{(z-z_0)}^{1+r_0}.\end{array}$$

Lemma 2 implies that

$$\begin{array}{ccc}& & {\displaystyle \frac{{(-p)}^r}{{(z-z_0)}^{\lceil r\rceil }}}{\left({\displaystyle \frac{z-z_0}{e^{z-z_0}-1}}\right)}^{\lceil r\rceil }={(-p)}^r\sum _{m\ge 0}{\displaystyle \frac{B_m^{(\lceil r\rceil )}}{m!}}{(z-z_0)}^{m-\lceil r\rceil }\hfill \\ & =& {(-p)}^r\left[{\displaystyle \sum _{m=0}^{\lceil r\rceil -1}}{\displaystyle \frac{s(\lceil r\rceil ,\lceil r\rceil -m)}{m!\left(\genfrac{}{}{0pt}{}{\lceil r\rceil -1}{m}\right)}}{(z-z_0)}^{m-\lceil r\rceil }+\sum _{j\ge 0}{a}_j{(z-z_0)}^j\right]\hfill \\ & =& {\displaystyle \sum _{j=1}^{\lceil r\rceil }}{\displaystyle \frac{(j-1)!{(-p)}^r}{(\lceil r\rceil -1)!}}s(\lceil r\rceil ,j){(z-z_0)}^{-j}+{\displaystyle \sum _{j=0}^{\infty }}{(-p)}^r{a}_j{(z-z_0)}^j.\hfill \end{array}$$

Therefore, from the poles method (Lemma 3) we have

$$\begin{array}{ccc}\hfill & & \left[z^n\right]f\left(z\right)\hfill \\ & \sim & \left[z^n\right]\left(\left[{(z-z_0)}^{r_0}+{\displaystyle \frac{r_0}{2}}{(z-z_0)}^{1+r_0}\right]{\displaystyle \sum _{j=1}^{\lceil r\rceil }}{\displaystyle \frac{(j-1)!{(-p)}^{r}s(\lceil r\rceil ,j)}{(\lceil r\rceil -1)!}}{(z-z_0)}^{-j}\right)\hfill \\ & \sim & p^r\left[z^n\right]{\displaystyle \sum _{j=1}^{\lceil r\rceil }}{\displaystyle \frac{(j-1)!\mathfrak{s}(\lceil r\rceil ,j)}{(\lceil r\rceil -1)!}}{(z_0-z)}^{r_0-j}\hfill \\ & & -{\displaystyle \frac{r_0{p}^r}{2}}\left[z^n\right]{\displaystyle \sum _{j=2}^{\lceil r\rceil }}{\displaystyle \frac{(j-1)!\mathfrak{s}(\lceil r\rceil ,j)}{(\lceil r\rceil -1)!}}{(z_0-z)}^{1+r_0-j}.\hfill \end{array}$$

(23)

Furthermore, since $r_0-j<0$ for $j\ge 1$,

$$\begin{array}{c}\hfill \left({\displaystyle \genfrac{}{}{0pt}{}{r_0-j}{n}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{-(j-r_0)}{n}}\right)={(-1)}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n+j-r_0-1}{n}}\right)={(-1)}^n{\displaystyle \frac{{\langle j-r_0\rangle }_n}{n!}}.\end{array}$$

Then,

$$\begin{array}{ccc}\hfill & & p^r\left[z^n\right]{\displaystyle \sum _{j=1}^{\lceil r\rceil }}{\displaystyle \frac{(j-1)!\mathfrak{s}(\lceil r\rceil ,j)}{(\lceil r\rceil -1)!}}{(z_0-z)}^{r_0-j}\hfill \\ & & =p^r{\displaystyle \sum _{j=1}^{\lceil r\rceil }}{\displaystyle \frac{\mathfrak{s}(\lceil r\rceil ,j)}{(\lceil r\rceil -1)!}}(j-1)!z_0^{r_0-j}\left({\displaystyle \genfrac{}{}{0pt}{}{r_0-j}{n}}\right){\displaystyle \frac{{(-1)}^n}{z_0^n}}\hfill \\ & & ={\displaystyle \frac{p^r}{n!(\lceil r\rceil -1)!}}{\displaystyle \sum _{j=1}^{\lceil r\rceil }}{\displaystyle \frac{(j-1)!{\langle j-r_0\rangle }_n}{{\left(\log (1/q)\right)}^{n+j-r_0}}}\mathfrak{s}(\lceil r\rceil ,j).\hfill \end{array}$$

(24)

Finally,

$$\begin{array}{ccc}\hfill & & -{\displaystyle \frac{r_0{p}^r}{2}}\left[z^n\right]{\displaystyle \sum _{j=2}^{\lceil r\rceil }}{\displaystyle \frac{(j-1)!\mathfrak{s}(\lceil r\rceil ,j)}{(\lceil r\rceil -1)!}}{(z_0-z)}^{1+r_0-j}\hfill \\ & & =-{\displaystyle \frac{r_0{p}^r}{2}}{\displaystyle \sum _{j=2}^{\lceil r\rceil }}{\displaystyle \frac{(j-1)!\mathfrak{s}(\lceil r\rceil ,j)}{(\lceil r\rceil -1)!}}{z}_0^{1+r_0-j}\left({\displaystyle \genfrac{}{}{0pt}{}{-(j-1-r_0)}{n}}\right){\displaystyle \frac{{(-1)}^n}{z_0^n}}\hfill \\ & & =-{\displaystyle \frac{r_0{p}^r}{2}}{\displaystyle \frac{1}{n!(\lceil r\rceil -1)!}}{\displaystyle \sum _{j=2}^{\lceil r\rceil }}{\displaystyle \frac{(j-1)!{\langle j-1-r_0\rangle }_n}{{\left(\log (1/q)\right)}^{n+j-1-r_0}}}\mathfrak{s}(\lceil r\rceil ,j).\hfill \end{array}$$

(25)

Since $\mathbf{E}{\left[NB(r,p)\right]}^n=n!\left[z^n\right]f\left(z\right)$, combining (23) to (25), we complete the proof of (20).

(2) Since the coefficients $\left[z^n\right]f_0\left(z\right)$ for $n\ge 2$ are all positive,

$$\begin{array}{c}\hfill f_0\left(z\right)={\displaystyle \sum _{i=2}^{\infty }}{\displaystyle \frac{q-q^i}{p^{i+1}}}{\displaystyle \frac{z^i}{i!}},0<p<1,q=1-p.\end{array}$$

The m.g.f. of $NB(r,p)-{\displaystyle \frac{q}{p}}r$ is

$$\begin{array}{ccc}\hfill \tilde{f}\left(z\right)& =& {\displaystyle \frac{p^r}{{(1-q{e}^z)}^r}}{e}^{-{\displaystyle \frac{q}{p}}rz}={({\displaystyle \frac{1}{p}}{e}^{{\displaystyle \frac{q}{p}}z}-{\displaystyle \frac{q}{p}}{e}^{{\displaystyle \frac{z}{p}}})}^{-r}={(1-f_0\left(z\right))}^{-r}\hfill \\ & =& {\displaystyle \sum _{k=0}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{-r}{k}}\right){(-f_0\left(z\right))}^k={\displaystyle \sum _{k=0}^{\infty }}\left({\displaystyle \genfrac{}{}{0pt}{}{r+k-1}{k}}\right){\left[{\displaystyle \sum _{i=2}^{\infty }}{\displaystyle \frac{q-q^i}{p^{i+1}}}{\displaystyle \frac{z^i}{i!}}\right]}^k.\hfill \end{array}$$

Hence, all $\left[z^n\right]\tilde{f}\left(z\right)$ for $n\ge 2$, i.e., all n-th central moments of the negative binomial distribution, are positive.

The n-th central moment of the negative binomial distribution is

$$\begin{array}{ccc}& & \left[{\displaystyle \frac{z^n}{n!}}\right]{\left({\displaystyle \frac{p}{1-q{e}^z}}\right)}^r{e}^{-{\displaystyle \frac{q}{p}}rz}\hfill \\ & =& \left[{\displaystyle \frac{z^n}{n!}}\right]\left[{\displaystyle \frac{{(-p)}^r{e}^{-\frac{q}{p}r{z}_0}}{{(z-z_0)}^{\lceil r\rceil }}}{(e^{z-z_0}-1)}^{r_0}·{\left({\displaystyle \frac{z-z_0}{e^{z-z_0}-1}}\right)}^{\lceil r\rceil }{e}^{-{\displaystyle \frac{q}{p}}r(z-z_0)}\right]\hfill \\ & \sim & \left[{\displaystyle \frac{z^n}{n!}}\right]\left[{(-p)}^r{q}^{{\displaystyle \frac{q}{p}}r}\left({(z-z_0)}^{r_0}+{\displaystyle \frac{r_0}{2}}{(z-z_0)}^{1+r_0}\right){\displaystyle \sum _{j=1}^{\lceil r\rceil }}{\displaystyle \frac{B_{\lceil r\rceil -j}^{(\lceil r\rceil )}(-\frac{q}{p}r)}{(\lceil r\rceil -j)!}}{(z-z_0)}^{-j}\right]\hfill \\ & \sim & \left[{\displaystyle \frac{z^n}{n!}}\right]{(-p)}^r{q}^{{\displaystyle \frac{q}{p}}r}{(z-z_0)}^{r_0-\lceil r\rceil }=p^r{q}^{{\displaystyle \frac{q}{p}}r}{\displaystyle \frac{{\langle r\rangle }_n}{{\left(\log (1/q)\right)}^{n+r}}}.\hfill \end{array}$$

(3) Notice that ${\langle \lceil r\rceil -r_0\rangle }_n={\langle r\rangle }_n$, and for $j<\lceil r\rceil$ we have

$$\begin{array}{c}\hfill {\langle j-r_0\rangle }_n=(j-r_0)\dots (j-r_0+n-1)=(r-i)(r-i+1)\dots (r+n-i-1),\end{array}$$

where $i=\lceil r\rceil -j\ge 1$.

Then for $j<\lceil r\rceil$,

$$\begin{array}{c}\hfill {\displaystyle \frac{{\langle j-r_0\rangle }_n}{{\langle \lceil r\rceil -r_0\rangle }_n}}={\displaystyle \frac{(r-(\lceil r\rceil -j\left)\right)\dots (r-1)}{(r+n-(\lceil r\rceil -j\left)\right)\dots (r+n-1)}}\to 0\mathrm{as}n\to +\infty .\end{array}$$

Since $\mathfrak{s}(k,k)=1$ for $k\ge 1$, result (20) yields the rough approximation as the following:

$$\begin{array}{c}\hfill \mathbf{E}{\left[NB(r,p)\right]}^n\sim p^r{\displaystyle \frac{{\langle r\rangle }_n}{{\left(\log (1/q)\right)}^{n+r}}},n\to +\infty ,\end{array}$$

which implies the asymptotic central-raw ratio (22).

The proof of Theorem 1 is complete. □

From an analytic perspective, the fundamental difference between Poisson-type distributions and the negative binomial distribution originates from the dominant singularity of their moment generating functions. The m.g.f. of $NB(r,p)$ possesses a finite pole at $z_0=\log (1/q)$, which acts as the dominant singularity controlling the asymptotic growth of high-order moments. This finite pole yields a nonvanishing limit ${lim}_{n\to \infty }{r}_n\left(NB\right)>0$. In contrast, the m.g.f. of the Poisson distribution is entire and has no finite singularity, leading to a vanishing ratio $r_n\left(\mathcal{P}\right)\to 0$. Thus, the location and nature of the dominant singularity determine the limiting behavior of the central-raw ratio.

Notice that $r_0=0$ in (20) if $r=k$ is positive integer. We have the following:

Corollary 1. 

The n-th raw moment of the Pascal distribution has the following asymptotic expansion:

$$\begin{array}{c}\hfill \mathbf{E}{\left[NB(k,p)\right]}^n\sim {\displaystyle \frac{p^k}{(k-1)!}}{\displaystyle \sum _{j=1}^k}{\displaystyle \frac{(n+j-1)!}{{\left(\log (1/q)\right)}^{n+j}}}\mathfrak{s}(k,j),n\to +\infty ,\end{array}$$

(26)

where $\mathfrak{s}(k,j)$ are the unsigned Stirling numbers of the first kind.

For $\frac{1}{2}\le p<1$, we have $0<q\le 0.5$. The minimal modulus simple pole $z_0=\log (1/q)$ still acts as the dominant singularity of the NB moment generating function, so the asymptotic limit $q^{qr/p}$ maintains an identical form; only the convergence speed slows down as p approaches 1, which can be verified via the numerical data in

Table 1. Raw moments of negative binomial distribution $NB(r,p)$.

n	2	3	4	5	6	12
Actual	4.6	46.1	663.0	12,495.2	291,421.5	909,837,797,955,762
Asymptotic	4.7	46.5	666.7	12,548.8	292,428.5	911,292,171,004,918
n	2	3	4	5	6	7	12
Actual	5.59	32.4	250.5	2429.3	28,306.7	385,234.6	1,026,756,899,758.5
Asymptotic	5.63	32.5	251.3	2435.1	28,363.6	385,898.9	1,027,792,467,013.4
n	2	3	4	5	6	7	8	12
Actual	0.88	1.82	4.72	14.7	52.7	215.3	981.1	1,035,323.8
Asymptotic	0.87	1.81	4.70	14.6	52.5	214.5	978.2	1,033,383.5

The values of $NB(0.2,0.2)$, $NB(0.95,0.4)$, and $NB(4.8,0.9)$ are shown in the 2nd, 4th, and 6th rows, respectively.

.

Remark 2. 

Asymptotic Formula (3) of H. S. Wilf is the special case $k=1$, $p=0.5$ in (26).

4. Asymptotic Moments for Gamma Distribution

The probability density function of the Gamma distribution $Ga(\alpha ,\beta )$ ($\alpha >0,\beta >0$) is defined as [2] (p. 109)

$$\begin{array}{c}\hfill p\left(x\right)={\displaystyle \frac{x^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right){\beta }^{\alpha }}}{e}^{-{\displaystyle \frac{x}{\beta }}},x>0.\end{array}$$

(27)

The m.g.f. of $Ga(\alpha ,\beta )$ is known to be

$$\begin{array}{c}\hfill f\left(t\right)={(1-\beta t)}^{-\alpha },\end{array}$$

(28)

and the n-th raw moment is

$$\begin{array}{c}\hfill \mathbf{E}{\left[Ga(\alpha ,\beta )\right]}^n={\beta }^n{\displaystyle \frac{\mathrm{\Gamma }(\alpha +n)}{\mathrm{\Gamma }\left(\alpha \right)}}.\end{array}$$

(29)

It is clear that the generating function of central moments of the Gamma distribution is

$$\begin{array}{c}\hfill \tilde{f}\left(t\right)={(1-\beta t)}^{-\alpha }{e}^{-\alpha \beta t}.\end{array}$$

(30)

Theorem 2. 

All the n-th ($n\ge 2$) central moments of the Gamma distribution are positive. The asymptotic central-raw ratio of the Gamma distribution is

$$\begin{array}{c}\hfill r_n\left(Ga(\alpha ,\beta )\right)=e^{-\alpha },n\to +\infty ,\end{array}$$

(31)

which is independent of the parameter β.

Proof. 

The proof leverages Darboux’s method to extract the leading singular contribution of the Gamma m.g.f. near its dominant singularity; scale invariance then simplifies the derivation to the standard shape parameter case $\beta =1$. Since $Ga(\alpha ,\beta )=\beta ·Ga(\alpha ,1)$ and the central-raw ratio $r_n$ is scale-invariant, it suffices to consider $\beta =1$. Then the moment generating function of $Ga(\alpha ,1)-\alpha$ is

$$\tilde{f}\left(t\right)={(1-t)}^{-\alpha }{e}^{-\alpha t}.$$

For $\alpha >0$, we expand $\tilde{f}\left(t\right)$ about $t_0=1$. Note that

$$e^{-\alpha t}=e^{-\alpha }+\alpha e^{-\alpha }(1-t)+\dots .$$

The Darboux method (Lemma 4) implies that

$$\begin{array}{ccc}\hfill \left[t^n\right]{(1-t)}^{-\alpha }{e}^{-\alpha t}& \sim & e^{-\alpha }\left({\displaystyle \genfrac{}{}{0pt}{}{n+\alpha -1}{n}}\right)+\alpha e^{-\alpha }\left({\displaystyle \genfrac{}{}{0pt}{}{n+\alpha -2}{n}}\right)\hfill \\ & \sim & e^{-\alpha }{\displaystyle \frac{\mathrm{\Gamma }(\alpha +n)}{n!\mathrm{\Gamma }\left(\alpha \right)}},n\to +\infty .\hfill \end{array}$$

Therefore,

$$\mathbf{E}{[Ga(\alpha ,1)-\alpha ]}^n=n!\left[t^n\right]\tilde{f}\left(t\right)\sim e^{-\alpha }{\displaystyle \frac{\mathrm{\Gamma }(\alpha +n)}{\mathrm{\Gamma }\left(\alpha \right)}},n\to +\infty .$$

By scale invariance, for general $\beta >0$,

$$\mathbf{E}{[Ga(\alpha ,\beta )-\alpha \beta ]}^n\sim e^{-\alpha }{\beta }^n{\displaystyle \frac{\mathrm{\Gamma }(\alpha +n)}{\mathrm{\Gamma }\left(\alpha \right)}}.$$

Combining with the raw moment $\mathbf{E}{\left[Ga(\alpha ,\beta )\right]}^n={\beta }^n\mathrm{\Gamma }(\alpha +n)/\mathrm{\Gamma }\left(\alpha \right)$ from (29), we obtain $r_n\left(Ga(\alpha ,\beta )\right)\sim e^{-\alpha }$. □

Similar singularity arguments apply to the Gamma distribution. Its moment generating function ${(1-\beta t)}^{-\alpha }$ has a finite dominant singularity at $t=1/\beta$, which governs the asymptotic expansion of high-order raw and central moments. The scale invariance of $r_n\left(Ga\right)$ follows directly from the homogeneous power form of the m.g.f., and the constant limit $e^{-\alpha }$ is a direct consequence of the residue contribution near this dominant singularity. This separates Gamma from normal and Poisson distributions, whose central-raw ratios vanish.

Remark 3. 

The n-th central-raw ratio of the Gamma distribution is

$$\begin{array}{c}\hfill r_n\left(Ga(\alpha ,\beta )\right)={\int }_0^{\infty }{\displaystyle \frac{x^{\alpha -1}{(x-\alpha )}^n{e}^{-x}}{\mathrm{\Gamma }(n+\alpha )}}dx,\end{array}$$

(32)

which is indeed independent of the parameter β. Theorem 2 gives the limit $e^{-\alpha }$ of the integral in (32) as $n\to +\infty$.

5. Numerical Results

In this section we give some simulations of accurate and asymptotic values for the raw moments $\mathbf{E}{\xi }^n$ and central-raw ratios $r_n\left(\xi \right)={\displaystyle \frac{\mathbf{E}{(\xi -\mathbf{E}\xi )}^n}{\mathbf{E}{\xi }^n}}$. All results are computed by Maple.

Table 1 includes some simulation results of the n-th raw moments of negative binomial distribution for different r and p. The actual values are evaluated by Formula (12), and the asymptotic values are evaluated by Formula (20).

Table 2. Raw moments of Pascal distribution $NB(k,p)$.

n	2	3	4	5	6	12
Actual	36	484	8676	194,404	5,227,236	28,168,941,794,178,300
Asymptotic	36.0	484.0	8676.0	194,404.0	5,227,236.0	28,168,941,794,178,300.0
n	2	3	4	5	6	7	12
Actual	7.3	35.8	217.1	1569.4	13,137.2	124,823.9	42,123,715,322.9
Asymptotic	7.3	35.8	217.1	1569.4	13,137.2	124,823.9	42,123,715,322.9
n	2	3	4	5	6	7	8	9	12
Actual	0.35	0.54	1.04	2.41	6.53	20.15	69.64	266.09	24,752.94
Asymptotic	0.35	0.54	1.04	2.41	6.53	20.15	69.64	266.08	24,752.97

The values of $NB(1,0.2)$, $NB(3,0.6)$, and $NB(5,0.95)$ are shown in the 2nd, 4th, and 6th rows, respectively.

gives some surprising simulation results of the n-th raw moments of Pascal distribution for different k and p. The actual values are evaluated by Formula (12), and the asymptotic values are evaluated by Formula (26).

Furthermore, for negative binomial distribution $NB(r,p)$, we assume that $r=1.7$ and $p=0.4$, so the asymptotic central-raw ratio is given to be $q^{{\displaystyle \frac{q}{p}}r}=0.2718226790$ by Formula (22). Since it is hard to compute the series ${\sum }_{i=0}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{0.7+i}{i}}\right)0.6^i{i}^n$ for large n, the n-th central-raw ratios are computed by the following truncated forms (see

Table 3. The n-th central-raw ratio of negative binomial distribution $NB(1.7,0.4)$.

n	100	300	500	800	1000
Ratios $r_n{\left(\xi \right)|}_T$	0.2742686378	0.2726452121	0.2723170727	0.2721319822	0.2720702022

The asymptotic central-raw ratio of $NB(1.7,0.4)$ is $0.2718226790$.

):

$$\begin{array}{c}\hfill r_n{\left(\xi \right)|}_T={\displaystyle \frac{{\sum }_{i=0}^{5n}\left(\genfrac{}{}{0pt}{}{0.7+i}{i}\right)0.6^i{(i-2.55)}^n}{{\sum }_{i=1}^{5n}\left(\genfrac{}{}{0pt}{}{0.7+i}{i}\right)0.6^i{i}^n}}.\end{array}$$

Finally, we consider the Gamma distribution $Ga(\alpha ,\beta )$ and assume $\alpha =2.5$ and $\beta =1$. From Formula (31) the asymptotic central-raw ratio of $Ga(2.5,1)$ is $e^{-\alpha }=0.08208499862$. The n-th central-raw ratio is computed by Formula (32). See

Table 4. The n-th central-raw ratio of Gamma distribution $Ga(2.5,1)$.

n	100	500	1000	1500	2000
Ratios $r_n\left(\xi \right)$	0.08513647745	0.08269956055	0.08239254827	0.08229009149	0.08223884070

The asymptotic central-raw ratio of $Ga(2.5,1)$ is $0.08208499862$.

.

6. Conclusions

It should be noted that the asymptotic formulas derived in Section 3 and Section 4 are obtained under the assumption $n\to \infty$; therefore, they are most accurate for large n. For small n, exact Formulas (12) and (29) should be used instead.

In this paper we use combinatorial and asymptotic methods, including the Lagrange inversion formula, generalized Bernoulli polynomials, poles method, and Darboux’s method, to derive the higher-order moments of the negative binomial distribution $NB(r,p)$ and the Gamma distribution $Ga(\alpha ,\beta )$. The n-th raw moment of $NB(r,p)$ is approximated by a sum of $\lceil r\rceil$ terms involving Stirling numbers of the first kind.

The numerical results confirm that the approximation formula for the raw moments of the Pascal distribution is very accurate. Moreover, the asymptotic central-raw ratios of the negative binomial and Gamma distributions are shown to be non-zero constants, which are different because of the fact that the asymptotic central-raw ratios of the binomial, Poisson and normal distributions are known to be infinitesimal.

Several directions for future research emerge from this work. First, the asymptotic classification framework based on $r_n$ could be extended to multivariate distributions and mixture models. Second, analogous high-order moment asymptotics for stable laws and heavy-tailed distributions deserve further investigation. Third, connections between central-raw ratios and information-theoretic measures such as entropy or cumulant generating functions may reveal new interpretations of distribution tail behavior. Fourth, the theoretical results here lay the groundwork for constructing formal hypothesis tests to distinguish Poisson and negative binomial count data via sample central-raw ratios. Finally, finite-sample correction terms could be developed to improve accuracy for moderate moment orders.