Stuttering Poisson Distribution (Poisson-stopped sum)

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What is the Stuttering Poisson Distribution?

 

The Stuttering Poisson Distribution (SPD), also called the Poisson-stopped sum or multiple Poisson, is a non-negative, discrete compound Poisson distribution that describes two or more events that happen in quick bursts. For example, the events might occur in groups or batches [1].

The distribution has the probability generating function (PGF):

Where a_i  (i = 1, 2, …) is the density of a positive discrete distribution.

Calculating Stuttering Poisson Distribution Probability

A general formula for calculating the probability of observing a demand equal to x is given by [2]:

For low demand (x = 1 or x = 2), the formula simplifies to the Poisson distribution.

The following table shows Poisson (λ = 2) and stuttering Poisson distribution (λ = 1 and  ρ = 5) probabilities and cumulative probabilities:

The term “stuttering” Poisson is mostly used in older literature, it does make an appearance in a few modern texts. Many modern authors call the distribution a Poisson-stopped sum or multiple Poisson. The SPD does have a variety of other names in the literature. For example, Cox [3] called the process a “cumulative process associated with a Poisson process.” It’s also referred to as:

• Composed distribution [4]
• Compound Poisson [5]. Note though, that the stuttering Poisson is actually a special case of compound Poisson distribution [6].
• Distribution par grappes [7],
• Poison distributions with events in clusters [8]
• Poisson power-series distribution [9]
• Pollaczek-Geiringer distribution.

The name “stuttering” Poisson distribution originated with Galliher et al. [10].  Patel [11] introduced the triple- and quadruple stuttering Poisson distributions.

Historical Notes on the Pollaczek-Geiringer Distribution

Another name for the stuttering Poisson distribution is the “The Pollaczek-Geiringer distribution.” It makes a sparse entry in 1958’s Index to the Distributions of Mathematical Statistics. National Bureau of Standards Report [12]:

The reference “[17] No. 9” refers to the obscure and seldom-referenced book A Summary of Known Distribution Functions, published in 1945[13].

Hilda Geiringer was born in Vienna in 1893. Her papers between 1923 and 1934 appeared under the hyphenated name Pollaczek-Geiringer due to her (brief) marriage to the statistician Felix Pollaczek (1892-1981). One paper was on “The Poisson distribution and the development of arbitrary distributions” which stirred up debate:

“…namely, expansions of a discrete distribution with an infinite number of values in a series in successive derivatives of the Poisson distribution with respect to the parameter. These expansions were first proposed by the Swedish astronomer C. L. W. Charlier (1862-1934) in 1905.” [14]

References

1. Zhang, H et al. (2012). Some Properties of the Generalized Stuttering Poisson Distribution and Its Applications. Studies in Mathematical Sciences. Vol. 5, No. 1, 2012, pp. [11–26] www.cscanada.net DOI: 10.3968/j.sms.1923845220120501.Z0697
2. Syntetos, A. & Boylan, J. (2021). Intermittent Demand Forecasting. Wiley.
3. Cox, D. R. (1962). Renewal Theory, Methuen, London
4. Janossy L. et al. (1950). 0n composed Poisson distributions, I, Acta. Math. Acad. ScL Hung., 1, pp. 209–224.
5. Feller, W. (1957). An Introduction to Probability Theory and Its Applications (2^nd ed.). Vol 1. New York. Wiley.
6. Willmot, G. (1986). Mixed compound Poisson distributions. Retrieved April 17, 2023 from: https://www.cambridge.org/core/services/aop-cambridge-core/content/view/EB000303D7A5230E79869B13CDEC04CE/S051503610001165Xa.pdf/mixed_compound_poisson_distributions.pdf
7. Thyrion, P. (1960). Note sur les distribution “par grappes.” Association Royal des Actuaires Belges Bulletin, 60. 49-66.
8. Castoldi, L. (1963). Poisson processes with events in clusters. Rendiconti del Seminaro della Facolta di Scienze della Universita di Cagliari, 33, 433-437.
9. KHATRI, C. G. & PATEL, I. R. (1961). Three classes of univariate discrete distributions. Biometrics, 17, 567-75.
10. GALLIHER, H. P., MORSE, P. M. and SIMOND, M. (1959). ‘ Dynamics of Two Classes of Continuous-Review Inventory Systems ‘, Opns. Res. 7, 362-383.
11. Patel, Y. C. (1976). Estimation of the parameters of the triple and quadruple stuttering Poisson distributions. Technometrics, 18,  67-73.
12. Haight, F. (1958). Index to the Distributions of Mathematical Statistics. National Bureau of Standards Report.
13. Haller, B. Verteilungsfunktionen und ihre Auszeichnung durch Funktionalgleichungen. Mitteilungen der Vereinigung schweizerischer Versicherungsmathematiker . 45 Band, Heft 1, 21 April 1945, pp. 97 – 163. Translated by R, E. Kalaba, and published by the RAND Corporation under the title A Summary of Known Distribution Functions, T – 27, 7 January 1953.
14. Siegmund-Schultze, R. Human Side of the Emancipation of Applied Mathematics at the University of Berlin During the 1920s. Historia Mathematica 20 (1993). 364-381.