Abstract
I simplify and note extensions of results of Shibu et al. Sankhy\(\overline{a}\), Ser. A 85 (2023a,b) concerning integral representations involving the probability generating function for inverse moments of positive discrete random variables, both univariate and multivariate.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Complements to Shibu et al. (2023a)
In a recent article in this journal, Shibu et al. (2023a) “derive exact expressions for inverse moments for any positive integer valued random variable" (X, with probability mass function [p.m.f.] \(p_1,p_2,...\)) using “an integral representation for inverse moments involving the probability generating function" (p.g.f., \(G(s)= {\mathbb {E}(s^X)=}\sum _{x=1}^\infty p_x s^x\)). For the first inverse moment, \(\mu _{-1} \equiv {\mathbb E}(1/X)\), they show that \(\mu _{-1} = \lim _{t \rightarrow \infty } \{tH(t)\}\) where \(H(t) \equiv \int _0^1 G(s^t)dt\). I would first like to note a different version of this result, namely,
To see this, note that the right-hand side equals
Note also that, for \(A>0\), by a similar argument,
Example expressions given by Shibu et al. for \(\mu _{-1}\) for geometric and negative binomial distributions are correct but both can be evaluated beyond the integral forms given. The first reduces to \(p(-\log p)/(1-p)\). Using
the second becomes
here, integer m is both the index and the starting point of the negative binomial distribution. When \(m=1\), the latter reduces to the former since then the sum vanishes.
Shibu et al. (2023a) also provide an expression for higher order integer inverse moments which is a limiting value of an expression involving H(t) and lower order integer inverse moments. To complement this with a corresponding direct integral expression, I note that Cressie et al. (1981) gave the formula
\(q>0\), where M(t) is the moment generating function given in terms of the p.g.f. by \(M(t) = G(e^{t})\). To see this, note that the right-hand side of (3) equals
In terms of the p.g.f., therefore, we have
Note that (4) reduces to (1) when \(q=1\).
Example 1
For the geometric distribution (starting at 1) with p.g.f. \(G(s) = ps/\{1-(1-p)s\}\), (4) shows that
While \(\mu _{-1} = p(-\log p)/(1-p)\), as noted above, (5) shows that for \(q=2,3,...\),
where \(\textrm{Li}_q(1-p)\) is the polylogarithm function as given, for example, in NIST (2024, Section 25.12). This route gives the integral representation of the polylogarithm function; directly, minor manipulation gives \(p/(1-p)\) times the series definition of the same function:
The second inverse moment, consequently, involves the dilogarithm function.
Note that the general results above apply to any positive discrete valued random variable, not just integer valued versions, via just a notational change to the indexing of x.
2 Related Expressions, Particularly Inverse Factorial Moments
“When studying discrete distributions, it is often advantageous to use the factorial moments" (Johnson et al., 2005, p.52). This is because many discrete distributions involve factorial terms, usually arising from combinatorial considerations. So, in a discrete, integer valued, context, it might be expected that inverse factorial moments can be easier to work with than inverse power moments. For \(\nu _{-n} \equiv \mathbb {E}(1/\{X(X+1)\cdots (X+n-1)\}= \mathbb {E}\{(X-1)!/(X+n-1)!\}\), \(n=1,2,...,\) I earlier gave a closely related formula with an unfortunate typographical error (the power \(n-1\) is missing in the theorem in Jones, 1987) which I here correct to
To confirm this, observe that the right-hand side is
equation (6) too reduces to (1) when \(n=1\). As shown in Jones (1987), (6) can also be proved by expanding \(1/\{X(X+1) \cdots (X+n-1)\}\) in partial fractions and using (2). And there is also an integral representation for \(\nu _{-n}\) rather like (3) for \(\mu _{-q}\) but involving the factorial moment generating function \(F_X(y)=\mathbb {E}\{(1+y)^X\}=G(1+y)\) rather than the moment generating function. This representation is
(the corollary in Jones, 1987). By way of proof, the right-hand side of (7) is
and the argument is completed as above.
Example 2
By using (6), the n’th inverse factorial moment of the geometric distribution is
where \(\,_2F_1(1,1;n+1;1-p)\) is a special case of the Gauss hypergeometric function (NIST, 2024, Section 15). Again, this approach directly yields an integral representation of the special function concerned; minor manipulation of the direct formula for the inverse factorial moment yields the usual series representation thereof.
More generally, other works concerning integral representations of inverse moments include Chao and Strawderman (1972); Adell et al. (1996) and Cressie and Borkent (1986).
3 Complement to Shibu et al. (2023b)
Shibu et al. (2023b) generalize the univariate results of Shibu et al. (2023a) to the multivariate case. Let \(X_1,...,X_k\) follow a positive integer valued multivariate distribution with p.m.f. \(p_{x_1 \ldots x_k}\), p.g.f. \(G(s_1,...,s_k) = \mathbb {E}(s_1^{X_1} \cdots s_k^{X_k})\) and inverse moments \(\mu _{-q_1,\ldots ,-q_k}= \mathbb {E}(X_1^{q_1} \cdots X_k^{q_k})\). Then, Shibu et al. (2023b) show that
together with a complicated expression for general \(\mu _{-q_1,\ldots ,-q_k}\) involving limits of multiple summations each involving lower order inverse moments. Formulas (1) and (4) generalise straightforwardly to the multivariate case, however. We have
The proof is no more difficult than in the univariate case: the right-hand side of (9) is
Example 3
Example 2 of Shibu et al. (2023b) concerns inverse moments of the Barbiero (2019) bivariate geometric distribution which has p.g.f. of the form
for \(a_j,b_j,c_j,\) \(j=1,...,4\), being certain simple functions of parameters\(0<\theta _1,\theta _2<1\) and \(-1\le \alpha \le 1\). Since in the bivariate case
it is easy to see that
This is simpler than the formula given by Shibu et al. (2023b).
References
Adell, J.A., de la Cal, J. and Peréz-Palomares, A. (1996). On the Cheney and Sharma operator. J. Math. Anal. Appl., 200, 663–679
Barbiero, A. (2019). A bivariate geometric distribution allowing for positive or negative correlation. Commun. Statist. Theory Meth., 48, 2842–2861.
Chao, M.T. and Strawderman, W.E. (1972) Negative moments of positive random variables. J. Amer. Statist. Assoc., 67, 429–431.
Cressie, N.A.C. and Borkent, M. (1986). The moment generating function has its moments. J. Statist. Planning Inference, 13, 337–344.
Cressie, N.A.C., Davis, A.S., Folks, J.L. and Policello, G.E. (1981). The moment-generating function and negative integer moments. Amer. Statist. 35, 148–150.
Johnson, N.L., Kemp, A.W. and Kotz, S. (2005). Univariate Discrete Distributions, 3rd edn. Wiley, Hoboken, NJ.
Jones, M.C. (1987). Inverse factorial moments. Statist. Probab. Lett. 6, 37–42.
NIST (2024). Digital Library of Mathematical Functions (eds: F.W.J. Olver, A.B. Olde Daalhuis, D.W. Lozier, B.I. Schneider, R.F. Boisvert, C.W. Clark, B.R. Miller, B.V. Saunders, H.S. Cohl and M.A. McClain). https://dlmf.nist.gov/, Release 1.2.0 of 2024-03-15.
Shibu, D.S., Irshad, M.R. and Nadarajah, S. (2023a). An integral representation for inverse moments. Sankhy\(\overline{a}\), Ser. A 85, 1394–1402.
Shibu, D.S., Irshad, M.R. and Nadarajah, S. (2023b). An integral representation for inverse moments of multivariate random variables. Stat. 12, e603.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
The author declares no conflict of interest nor was any funding received to assist with the preparation of this manuscript. He is grateful to the referees for suggesting several improvements to the article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Jones, M.C. On Integral Representations Involving the Probability Generating Function for Inverse Moments of Positive Discrete Random Variables. Sankhya A (2024). https://doi.org/10.1007/s13171-024-00360-y
Received:
Published:
DOI: https://doi.org/10.1007/s13171-024-00360-y
Keywords
- Geometric distribution
- Inverse factorial moments
- Multivariate inverse moments
- Negative binomial distribution
- Negative moments