Stuttering Poisson Distribution (Poisson-stopped sum)

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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 Px is the probability a discrete random variable will take on the value x.

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 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

The Pollaczek-Geiringer distribution, another name for the stuttering Poisson distribution, 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-reference 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] Huiming, Z. 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 (2nd 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.