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Handy formulas for binomial moments
Volume 12, Issue 1 (2025), pp. 27–41
Maciej Skorski ORCID icon link to view author Maciej Skorski details  

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https://doi.org/10.15559/24-VMSTA260
Pub. online: 30 July 2024      Type: Research Article      Open accessOpen Access

Received
10 April 2024
Revised
19 June 2024
Accepted
19 June 2024
Published
30 July 2024

Abstract

Despite the relevance of the binomial distribution for probability theory and applied statistical inference, its higher-order moments are poorly understood. The existing formulas are either not general enough, or not structured and simplified enough for intended applications.
This paper introduces novel formulas for binomial moments in the form of polynomials in the variance rather than in the success probability. The obtained formulas are arguably better structured, simpler and superior in their numerical properties compared to prior works. In addition, the paper presents algorithms to derive these formulas along with working implementation in Python’s symbolic algebra package.
The novel approach is a combinatorial argument coupled with clever algebraic simplifications which rely on symmetrization theory. As an interesting byproduct asymptotically sharp estimates for central binomial moments are established, improving upon previously known partial results.

References

[1] 
Agresti, A., Coull, B.A.: Approximate is better than “exact” for interval estimation of binomial proportions. The American Statistician 52(2), 119–126 (1998) MR1628435. https://doi.org/10.2307/2685469
[2] 
Balakrishnan, N., Johnson, N.L., Kotz, S.: A note on relationships between moments, central moments and cumulants from multivariate distributions. Statistics & probability letters 39(1), 49–54 (1998) MR1649335. https://doi.org/10.1016/S0167-7152(98)00027-3
[3] 
Bényi, Á., Manago, S.M.: A recursive formula for moments of a binomial distribution. The College Mathematics Journal 36(1), 68–72 (2005)
[4] 
Boyadzhiev, K.N.: Close encounters with the Stirling numbers of the second kind. Mathematics Magazine 85(4), 252–266 (2012) MR2993818. https://doi.org/10.4169/math.mag.85.4.252
[5] 
Buchberger, B.: Gröbner bases: A short introduction for systems theorists. In: International Conference on Computer Aided Systems Theory, pp. 1–19 (2001). Springer MR1868810. https://doi.org/10.1023/A:1011949421611
[6] 
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. The MIT Press (2001) MR1848805
[7] 
Daniel Paulino, C., Soares, P., Neuhaus, J.: Binomial regression with misclassification. Biometrics 59(3), 670–675 (2003) MR2004272. https://doi.org/10.1111/1541-0420.00077
[8] 
De Moivre, A.: Approximatio ad summum terminorum binomii seriem expansi. Proceedings of the Royal Society (1733)
[9] 
Fraas, J.W., Newman, I.: A binomial test of model fit. Structural Equation Modeling: A Multidisciplinary Journal 1(3), 268–273 (1994)
[10] 
Gaudry, P., Schost, É., Thiéry, N.M.: Evaluation properties of symmetric polynomials. International Journal of Algebra and Computation 16(03), 505–523 (2006) MR2241620. https://doi.org/10.1142/S0218196706003128
[11] 
Goodson, M.: Most winning A/B test results are illusory. Whitepaper, Qubit (2014)
[12] 
Graham, R.L., Knuth, D.E., Patashnik, O., Liu, S.: Concrete mathematics: a foundation for computer science. Computers in Physics 3(5), 106–107 (1989) MR1001562
[13] 
Griffiths, M.: Raw and central moments of binomial random variables via Stirling numbers. International Journal of Mathematical Education in Science and Technology 44(2), 264–272 (2013) MR3172571. https://doi.org/10.1080/0020739X.2012.678899
[14] 
Joarder, A.H., Mahmood, M.: Classroom note: An inductive derivation of Stirling numbers of the second kind and their applications in statistics. Advances in Decision Sciences 1(2), 151–157 (1997) MR1609730. https://doi.org/10.1155/S1173912697000138
[15] 
Knoblauch, A.: Closed-form expressions for the moments of the binomial probability distribution. SIAM Journal on Applied Mathematics 69(1), 197–204 (2008) MR2447945. https://doi.org/10.1137/070700024
[16] 
Laplace, P.-S.: Mémoire sur les approximations des formules qui sont fonctions de très grands nombres et sur leur applications aux probabilités. Memoires de l’Academie des Sciences de Paris (1810)
[17] 
Little, R.J.: Testing the equality of two independent binomial proportions. The American Statistician 43(4), 283–288 (1989)
[18] 
Meurer, A., Smith, C.P., Paprocki, M., Čertík, O., Kirpichev, S.B., Rocklin, M., Kumar, A., Ivanov, S., Moore, J.K., Singh, S., Rathnayake, T., Vig, S., Granger, B.E., Muller, R.P., Bonazzi, F., Gupta, H., Vats, S., Johansson, F., Pedregosa, F., Curry, M.J., Terrel, A.R., Roučka, v., Saboo, A., Fernando, I., Kulal, S., Cimrman, R., Scopatz, A.: Sympy: symbolic computing in python. PeerJ Computer Science 3, 103 (2017). https://doi.org/10.7717/peerj-cs.103
[19] 
Nguyen, D.: A probabilistic approach to the moments of binomial random variables and application. The American Statistician, 75(1), 101–103 (2019). MR4203486. https://doi.org/10.1080/00031305.2019.1679257
[20] 
Patel, J.K., Read, C.B.: Handbook of the Normal Distribution, Second Edition. Statistics: A Series of Textbooks and Monographs. Taylor & Francis (1996). https://books.google.fr/books?id=zoVLF0VF9UYC MR0664762
[21] 
Potts, R.: Note on the factorial moments of standard distributions. Australian Journal of Physics 6(4), 498–499 (1953)
[22] 
Prasolov, V.V.: Polynomials, volume 11 of Algorithms and Computation in Mathematics. Springer (2004) MR2082772. https://doi.org/10.1007/978-3-642-03980-5
[23] 
Skorski, M.: Johnson-Lindenstrauss transforms with best confidence. In: Belkin, M., Kpotufe, S. (eds.) Proceedings of Thirty Fourth Conference on Learning Theory. Proceedings of Machine Learning Research, vol. 134, pp. 3989–4007. PMLR (2021)
[24] 
Skorski, M.: Bernstein-type bounds for Beta distribution. Modern Stochastics: Theory and Applications 10(2), 211–228 (2023) MR4573679
[25] 
Uspensky, J.V.: Introduction to Mathematical Probability. McGraw-Hill Book Company (1937). https://books.google.at/books?id=aeRQAAAAMAAJ
[26] 
Yang, J., Liu, Y., Liu, Z., Zhu, X., Zhang, X.: A new feature selection algorithm based on binomial hypothesis testing for spam filtering. Knowledge-Based Systems 24(6), 904–914 (2011)
[27] 
Young-Xu, Y., Chan, K.A.: Pooling overdispersed binomial data to estimate event rate. BMC medical research methodology 8(1), 58 (2008)
[28] 
Zhou, J.: Introduction to symmetric polynomials and symmetric functions. Lecture Notes for Course at Tsinghua University, available at http://cms.zju.edu.cn/course/cn/Symmetric.pdf (2003)
[29] 
Zolotukhin, A., Nagaev, S., Chebotarev, V.: On a bound of the absolute constant in the Berry–Esseen inequality for i.i.d. Bernoulli random variables. Modern Stochastics: Theory and Applications 5(3), 385–410 (2018). doi:https://doi.org/10.15559/18-VMSTA113 MR3868547. https://doi.org/10.15559/18-vmsta113

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Keywords
Binomial distribution high-order moments moment asymptotics symbolic algebra

MSC2010
60E05

Funding
This research was partly done during the stay at the University of Luxembourg. The author acknowledges funding from the FNR grant C17/IS/11613923.

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