Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-18T12:30:28.812Z Has data issue: false hasContentIssue false
  • English
  • Français

On fractional linear bounds for probability generating functions

Published online by Cambridge University Press:  24 August 2016

Claude Lefevre ,
Marc Hallin  and
Prakash Narayan
Claude Lefevre*
Affiliation:
Université Libre de Bruxelles
Marc Hallin*
Affiliation:
Université Libre de Bruxelles
Prakash Narayan
Affiliation:
Monash University
*
∗Postal address: Université Libre de Bruxelles, Institut de Statistique, Campus Plaine, C.P.210, Boulevard du Triomphe, B-1050 Bruxelles, Belgium.
∗Postal address: Université Libre de Bruxelles, Institut de Statistique, Campus Plaine, C.P.210, Boulevard du Triomphe, B-1050 Bruxelles, Belgium.
Get access Rights & Permissions [Opens in a new window]

Abstract

The best upper and lower bounds for any probability generating function with mean m and finite variance are derived within the family of fractional linear functions with mean m. These are often intractable and simpler bounds, more useful for practical purposes, are then constructed. Direct applications in branching and epidemic theories are briefly presented; a slight improvement of the bounds is obtained for infinitely divisible distributions.

Keywords

FRACTIONAL LINEAR FUNCTIONINFINITELY DIVISIBLE DISTRIBUTIONBOUNDSBRANCHING PROCESSS–I–S INFECTIOUS DISEASE
Type
Research Paper
Information
Journal of Applied Probability , Volume 23 , Issue 4 , December 1986 , pp. 904 - 913
Copyright
Copyright © Applied Probability Trust 1986 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

∗∗

Present address: American Health and Life Insurance Company, 300 St Paul Place, BSP 14B, Baltimore, MD 21202, USA.

References

Ades, M., Dion, J.-P., Labelle, G. and Nanthi, K. (1982) Recurrence formula and the maximum likelihood estimation of the age in a simple branching process. J. Appl. Prob. 19, 776–784.Google Scholar
Agresti, A. (1974) Bounds on the extinction time distribution of a branching process. Adv. Appl. Prob. 6, 322–335.CrossRefGoogle Scholar
Agresti, A. (1975) On the extinction times of varying and random environment branching processes. J. Appl. Prob. 12, 39–46.CrossRefGoogle Scholar
Brook, D. (1966) Bounds for moment generating functions and for extinction probabilities. J. Appl. Prob. 3, 171–178.CrossRefGoogle Scholar
Daley, D. J. and Narayan, P. (1980) Series expansions of probability generating functions and bounds for the extinction probability of a branching process. J. Appl. Prob. 17, 939–947.CrossRefGoogle Scholar
Feller, W. (1967) An Introduction to Probability Theory and its Applications, Vol. 1, 3rd edn. Wiley, New York.Google Scholar
Fujimagari, T. (1980) On the extinction time distribution of a branching process in varying environments. Adv. Appl. Prob. 12, 350–366.CrossRefGoogle Scholar
Heyde, C. C. and Schuh, H.-J. (1978) Uniform bounding of probability generating functions and the evolution of reproduction rates in birds. J. Appl. Prob. 15, 243–250.CrossRefGoogle Scholar
Hwang, T.-Y. and Wang, N.-S. (1979) On best fractional linear generating function bounds. J. Appl. Prob. 16, 449–453.CrossRefGoogle Scholar
Lefevre, C. and Malice, M.-P. (1986) A discrete time model for a S–I–S infectious disease with a random number of contacts between individuals. Math. Modelling 7. To appear.Google Scholar
Narayan, P. (1981) On bounds for probability generating functions. Austral. J. Statist. 23, 80–90.CrossRefGoogle Scholar
Quine, M. P. (1976) Bounds for the extinction probability of a simple branching process. J. Appl. Prob. 13, 9–16.CrossRefGoogle Scholar
Turnbull, B. W. (1973) Inequalities for branching processes. Ann. Prob. 1, 457–474.Google Scholar