In this paper, we introduced a parametric family of discrete probability distributions that had a simpler probabilistic construction than the classical Lagrangian family of distributions, obtained by composing some counting distributions with bounded support with the natural exponential family. This new family included, as a special case, the natural exponential family of distributions and consequently some of the well-known classical discrete distributions in the statistical literature, such as the Poisson, binomial, negative binomial, Hyper-Poisson, and others. A useful generalization of this family, encompassing quasi-binomial and generalized Poisson distributions, was studied. A few stochastic interpretations of these families were described. An extension of the proposed family that induced a dispersion parameter was also discussed. Finally, an extensive comparison with several standard count-data models illustrated the flexibility and competitiveness of the proposed family.
Citation: Héctor W. Gómez, Emilio Gómez-Déniz, Diego I. Gallardo. On the composition of discrete distributions[J]. AIMS Mathematics, 2026, 11(8): 24120-24136. doi: 10.3934/math.2026973
In this paper, we introduced a parametric family of discrete probability distributions that had a simpler probabilistic construction than the classical Lagrangian family of distributions, obtained by composing some counting distributions with bounded support with the natural exponential family. This new family included, as a special case, the natural exponential family of distributions and consequently some of the well-known classical discrete distributions in the statistical literature, such as the Poisson, binomial, negative binomial, Hyper-Poisson, and others. A useful generalization of this family, encompassing quasi-binomial and generalized Poisson distributions, was studied. A few stochastic interpretations of these families were described. An extension of the proposed family that induced a dispersion parameter was also discussed. Finally, an extensive comparison with several standard count-data models illustrated the flexibility and competitiveness of the proposed family.
| [1] |
P. Consul, L. Shenton, Use of Lagrange expansion for generating generalized probability distributions, SIAM J. Appl. Math., 23 (1972), 239–248. https://doi.org/10.1137/0123026 doi: 10.1137/0123026
|
| [2] |
P. Consul, G. Jain, A generalization of the Poisson distribution, Technometrics, 15 (1973), 791–799. https://doi.org/10.1080/00401706.1973.10489112 doi: 10.1080/00401706.1973.10489112
|
| [3] |
G. Jain, P. Consul, A generalized negative binomial distribution, SIAM J. Appl. Math., 21 (1971), 501–513. https://doi.org/10.1137/0121056 doi: 10.1137/0121056
|
| [4] | P. Consul, Generalized Poisson Distributions. Properties and Applications, Marcel Dekker, Inc., New York, 1989. |
| [5] |
G. Jain, R. Gupta, A logarithmic series type distribution, Trab. Estad. Invest. Oper., 24 (1973), 99-105. https://doi.org/10.1007/BF03013757 doi: 10.1007/BF03013757
|
| [6] |
P. Consul, Relation of modified power series distributions to Lagrangian probability distributions, Commun. Stat.-Theor. Meth., 10 (1981), 2039–2046. https://doi.org/10.1080/03610928108828171 doi: 10.1080/03610928108828171
|
| [7] | N. Johnson, A. Kemp, S. Kotz, Univariate Discrete Distributions, John Wiley, INC, 2005. https://doi.org/10.1002/0471715816 |
| [8] | P. Consul, F. Famoye, Lagrangian Probability Distributions, Birkhäuser Boston, 2006. |
| [9] |
I. Tomoaki, S.-H. Ong, A Lagrangian non-central negative binomial distribution of the first kind, Commun. Stat.-Theor. Meth., 42 (2013), 466–477. https://doi.org/10.1080/03610926.2011.582564 doi: 10.1080/03610926.2011.582564
|
| [10] | K. M. Sakthivel, J. Rajkumar, A. Arul, Generating Haight-Poisson distribution using Lagrange expansion, Journal of Information and Computational Science, 9 (2019), 930–935. |
| [11] |
M. Rasheed, C. Chesneau, D. Santhamani, M. Monisha, R. Maya, Lagrangian zero truncated Poisson distribution: properties, regression model and applications, Symmetry, 14 (2022), 177. https://doi.org/10.3390/sym14091775 doi: 10.3390/sym14091775
|
| [12] |
K. G. Janardan, B. R. Rao, Lagrange distributions of the second kind and weighted distributions, SIAM J. Appl. Math., 43 (1983), 501–513. https://doi.org/10.1137/0143021 doi: 10.1137/0143021
|
| [13] |
K. G. Janardan, A wider class of Lagrange distributions of the second kind, Commun. Stat.-Theor. Meth., 26 (1997), 2087–2091. https://doi.org/10.1080/03610929708832035 doi: 10.1080/03610929708832035
|
| [14] | S. Li, F. Famoye, C. Lee, On the generalized Lagrangian probability distributions, Journal of Probability and Statistical Science, 8 (2010), 113–123. |
| [15] |
W. Jewell, Credible means are exact Bayesian for exponential families, ASTIN Bulletin, 8 (1974), 77–90. https://doi.org/10.1017/S0515036100009193 doi: 10.1017/S0515036100009193
|
| [16] |
P. Diaconis, D. Ylvisaker, Conjugate priors for exponential families, Ann. Statist., 7 (1979), 269–281. https://doi.org/10.1214/aos/1176344611 doi: 10.1214/aos/1176344611
|
| [17] | S. MacEachern, A characterization of some conjugate prior distributions for exponential families, Scand. J. Stat., 20 (1993), 77–82. |
| [18] | O. Barndorff-Nielsen, Information and Exponential Families in Statistical Theory, John Wiley & Sons, Chichester, 1978. |
| [19] | L. Brown, Fundamentals of Statistical Exponential Families, Institute of Mathematical Statistics, Hayward, 1986. |
| [20] | E. Lehmann, G. Casella, Theory of Point Estimation, Springer, New York, 1998. |
| [21] | P. Bickel, K. Doksum, Mathematical Statistics: Basic Ideas and Selected Topics, Vol. I, Pearson Prentice Hall, Upper Saddle River, 2006. |
| [22] |
P. Consul, L. Shenton, Some interesting properties of Lagrangian distributions, Commun. Stat.-Theor. Meth., 2 (1973), 263–272. https://doi.org/10.1080/03610927308827073 doi: 10.1080/03610927308827073
|
| [23] | R. Conway, W. Maxwell, A queuing model with state dependent service rates, Journal of Industrial Engineering, 12 (1962), 132–136. |
| [24] | M. Abramowitz, I. A. Stegun, (Eds.) Handbook of mathematical functions with formulas, graphs, and mathematical tables (Vol. 55), US Government Printing Office, 1964. |
| [25] | J. Gurland, R. Tripathi, Estimation of parameters on some extensions of the Katz family of discrete distributions involving hypergeometric functions, A Modern Course on Statistical Distributions in Scientific Work: Volume 1—Models and Structures Proceedings of the NATO Advanced Study Institute held at the University of Calgagry, Calgary, Alberta, Canada July 29–August 10, 1974, (1975), 59–82. https://doi.org/10.1007/978-94-010-1842-5_6 |
| [26] |
G. E. Bardwell, E. L. Crow, A two–parameter family of hyper–Poisson distributions, J. Am. Stat. Assoc., 59 (1964), 133–141. https://doi.org/10.1080/01621459.1964.10480706 doi: 10.1080/01621459.1964.10480706
|
| [27] |
S. Bhattacharya, Confluent hypergeometric distributions of discrete and continuous type with applications to accident proneness, Calcutta Statist. Ass. Bull., 15 (1966), 20–31. https://doi.org/10.1177/0008068319660103 doi: 10.1177/0008068319660103
|
| [28] |
P. C. Consul, On some properties and applications of quasi-binomial distribution, Commun. Stat.-Theor. Meth., 19 (1990), 477–504. https://doi.org/10.1080/03610929008830214 doi: 10.1080/03610929008830214
|
| [29] |
Y. Hu, X. Peng, T. Li, H. Guo, On the Poisson approximation to photon distribution for faint lasers, Phys. Lett. A, 367 (2007), 173–176. https://doi.org/10.1016/j.physleta.2007.03.004 doi: 10.1016/j.physleta.2007.03.004
|
| [30] | P. C. Consul, A simple urn model dependent upon predetermined strategy, Sankhya$\bar{a}$: The Indian Journal of Statistics, Series B, 36 (1974), 391–399. |
| [31] |
J. Rodrigues, M. De Castro, N. Balakrishnan, V. G. Cancho, Destructive weighted Poisson cure rate models, Lifetime Data Analysis, 17 (2011), 333–346. https://doi.org/10.1007/s10985-010-9189-2 doi: 10.1007/s10985-010-9189-2
|
| [32] |
P. Borges, J. Rodrigues, N. Balakrishnan, Correlated destructive generalized power series cure rate models and associated inference with an application to a cutaneous melanoma data, Comput. Stat. Data Anal., 56 (2012), 1703–1713. https://doi.org/10.1016/j.csda.2011.10.013 doi: 10.1016/j.csda.2011.10.013
|
| [33] |
D. I. Gallardo, H. Bolfarine, A. C. Pedroso-de-Lima, Destructive weighted Poisson cure rate models with bivariate random effects: Classical and Bayesian approaches, Comput. Stat. Data Anal., 98 (2016), 31–45. https://doi.org/10.1016/j.csda.2015.12.006 doi: 10.1016/j.csda.2015.12.006
|
| [34] |
D. I. Gallardo, H. Bolfarine, A. C. Pedroso-de-Lima, J. S. Romeo, Destructive power series long-term survival model with complex activation schemes, Stat. Interface, 12 (2019), 561–571. https://doi.org/10.4310/SII.2019.v12.n4.a6 doi: 10.4310/SII.2019.v12.n4.a6
|
| [35] |
J. Nelder, R. Wedderburn, Generalized linear models, J. R. Stat. Soc. A Stat., 135 (1972), 370–384. https://doi.org/10.2307/2344614 doi: 10.2307/2344614
|
| [36] | B. Jørgensen, The Theory of Dispersion Models, Chapman and Hall, London, 1997. |
| [37] |
E. Gómez-Déniz, A new discrete distribution: Properties and applications in medical care, J. Appl. Stat., 40 (2013), 2760–2770. https://doi.org/10.1080/02664763.2013.827161 doi: 10.1080/02664763.2013.827161
|
| [38] | Wolfram Research, Mathematica, Version 14.0, Wolfram Research, Inc., Champaign, Illinois, 2026. |
| [39] | R Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, 2026. |
| [40] | MathWorks, MATLAB Release 2026a, The MathWorks, Inc., Natick, Massachusetts, 2026. |
| [41] |
G. Willmot, The Poisson-inverse Gaussian distribution as an alternative to the negative binomial, Scand. Actuar. J., 1987 (1987), 113–127. https://doi.org/10.1080/03461238.1987.10413823 doi: 10.1080/03461238.1987.10413823
|
Héctor W. Gómez, Emilio Gómez-Déniz, Diego I. Gallardo. On the composition of discrete distributions[J]. AIMS Mathematics, 2026, 11(8): 24120-24136. doi: 10.3934/math.2026973
DownLoad: