Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T14:35:44.781Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Numerical Calculation of the Distribution Functions Defining Some Compound Poisson Processes*

Published online by Cambridge University Press:  29 August 2014

Carl Philipson
Carl Philipson*
Affiliation:
Stockholm, Sweden
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

1. The comfound Poisson process in the wide sense is defined as a process for which the probability distribution of the number i of changes in the random function attached to the process, while the parameter passes from o to a fixed value τ of the parameter measured on a suitable scale, is given by the Laplace-Stieltjes integral

where U(ν, τ) for a fixed value of τ defines the distribution of ν. U(ν, τ) is called the risk distribution and is either τ-independent or, dependent on ν, τ.

2. The compound Poisson process in the narrow sense is defined as a process for which the probability distribution of the number of changes can be written in the form of (I) with a τ-independent risk distribution.

In their general form these processes have been analyzed by Ove Lundberg (1940). For such processes the following relation holds for the probability of i changes in the interval ο to τ, P̅i (τ) say

this relation does not hold for processes with τ-dependent risk distribution. Hofmann (1955) has introduced a sub-set of the processes concerned in this section for which the probability for non-occurrence of a change in the interval o to τ is defined as a solution of the differential equation

and ϰ ≥ o; the solutions may be written in the form where η is independent of and of two alternative forms one for ϰ = I and one for other values of ϰ. The probabilities for i changes in the interval o to τ in the processes defined by the solutions of Hofmann's equation are derived by Leibniz's formula, and are designated by and, in this paper, called Hofmann probabilities.

Type
Papers
Information
ASTIN Bulletin: The Journal of the IAA , Volume 3 , Issue 1 , December 1963 , pp. 20 - 42
Copyright
Copyright © International Actuarial Association 1963

Footnotes

*

Presented to the Colloquium 1962 in Juan-les-Pins

References

LITERATURE CITED

Almer, B., 1957, Risk Analysis in Theory and Practical Statistics, Trans. XVth Int. Congr. Act., New York 1957, II, PP. 314–370.Google Scholar
Almer, B., 1962, Individual Risk Theory and Risk Statistics as applied to Fire Insurance, Report to the ASTIN Colloquium 1962, Juan-les-Pins.Google Scholar
Ammeter, H., 1948, A Generalization of the Collective Theory of Risk in Regard to Fluctuating Basic-Probabilities, Skand. Akt. Tidskr. 1948: 3–4, pp. 171–198. (For further references see Philipson, 1961 (b).)Google Scholar
Campagne, C., 1962, Sur les événements en chaîne et la distribution binomiale négative généralisée, Rapport à l'ASTIN Colloquium 1962, Juan-les-Pins.Google Scholar
Cramér, H., 1955, Collective Risk Theory, A Survey from the Point of View of the Theory of Stochastic Processes, Stockholm, Skandia Jubilee Volume, 1955, 92 pages. (For further references see Philipson, 1961 (b).)Google Scholar
Delaporte, P., 1959, Quelques problèmes de statistiques mathématique posés par l'assurance automobile et le bonus pour non sinistre, Bull. trim, de I'Inst. Act. Franҁais, 65, pp. 87–102.Google Scholar
Delaporte, P., 1960, Un problème de tarification de l'assurance accidents d'automobiles examiné par la statistique mathématique, Trans. XVIth Int. Congr. Act., Brussels 1960, pp. 121–135.Google Scholar
Esscher, Fr., 1932, On the Probability Function in the Collective Theory of Risk, Skand. Akt. Tidskr. 1932, p. 175.Google Scholar
Esscher, Fr., 1961–1962, Memoranda on the Numerical Calculation of the Distribution Function in the Theory of Risk, Stockholm (unpublished).Google Scholar
Fisher, R. and SirYates, F., 1938, Statistical Tables for Biological, Agricultural and Medical Research, London, Edinburgh.Google Scholar
Hofmann, M., 1955, Über zusammengesetzte Poisson-Prozesse und ihre Anwendungen in der Unfallsversicherung, Mitteil. Verein. Schweiz. Vers. Math., Vol. 55–3.Google Scholar
Lundberg, Filip, 1934, On tne Numerical Application of the Collective Risk Theory, Stockholm, De Förenade Jubilee Volume, 1934. (For further references see Philipson, 1961 (b).)Google Scholar
Lundberg, Ove, 1940, On Random Processes and Their Application to Sickness and Accident Statistics, Inaug. Diss. Uppsala, 172 pages.Google Scholar
Matérn, B., 1960, Spatial Variation, Stochastic Models and Their Applications to some Problems in Forest Surveys and other Sampling Investigations, Inaug. diss, Bull. Swed. Sta. Inst. Forestry Research, 49, 144 pages.Google Scholar
Pearson, E. S. and Hartley, H. O., 1954, Biometrika Tables for Statisticians, Vol. I., Cambridge Univ. Press.Google Scholar
Philipson, C., 1960 (a) Note on the Application of Compound Poisson Processes to Sickness and Accident Statistics, The ASTIN Bull., Vol. I Part IV, pp. 225–237.Google Scholar
Philipson, C., 1961 (b) Note on the Background to the Subject: Theory of Risk, Fundamental Mathematics and Applications, The ASTIN Bull. Vol. I Part V, pp. 256–264. (based on Report I to the Rattvik Colloquium of ASTIN).CrossRefGoogle Scholar
Philipson, C., 1961 (c) An Extension of the Models usually applied to the Theory of Risk, Report II to the Rättvik Colloquium of ASTIN Skand, Akt. Tidskr. 1961; 3–4 p. 223–239.Google Scholar
Philipson, C., 1961 (d) A Generalization of the Stochastic Processes commonly applied to the Theory of Causalty Risk, Bull, de l'Inst. Int. de Statistique, 33e Session, Paris.Google Scholar
Segerdahl, C. O., 1955, When does Ruin occur in the Collective Theory of Risk, Skand. Akt. Tidskr. 1955: 1–2, pp. 22–36. (For further references see the paper quoted).Google Scholar
Slater, L. J., 1960, Confluent Hypergeometric Functions, Cambr. Univ. Press, 243 pages.Google Scholar
Thyrion, P., 1960, Note sur les distribution “par grappes”, Bull. Ass. Roy Act. Belg., 1960: 60, pp. 49–66.Google Scholar