Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-12T22:09:36.791Z Has data issue: false hasContentIssue false
  • English
  • Français

Recurrence formula and the maximum likelihood estimation of the age in a simple branching process

Published online by Cambridge University Press:  14 July 2016

M. Adès ,
J.-P. Dion ,
G. Labelle  and
K. Nanthi
M. Adès*
Affiliation:
Université du Québec à Montréal
J.-P. Dion*
Affiliation:
Université du Québec à Montréal
G. Labelle*
Affiliation:
Université du Québec à Montréal
K. Nanthi*
Affiliation:
Presidency College, Madras
*
∗ Postal address: Université du Québec àMontréal, Case Postale 8888, Succ. A, Montréal, P.Q. H3C 3P8, Canada.
∗ Postal address: Université du Québec àMontréal, Case Postale 8888, Succ. A, Montréal, P.Q. H3C 3P8, Canada.
∗ Postal address: Université du Québec àMontréal, Case Postale 8888, Succ. A, Montréal, P.Q. H3C 3P8, Canada.
∗∗ Postal address: Presidency College, Madras-5, India.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we consider a Bienaymé– Galton–Watson process {Xn ; n ≧ 0; Xn = 1} and develop a recurrence formula for P(Xn = k), k = 1, 2, ···. The problem of obtaining the maximum likelihood estimate of the age of the process when p 0 = 0 is discussed. Furthermore the maximum likelihood estimate of the age of the process when the offspring distribution is negative binomial (p 0 ≠ 0) is obtained, and a comparison with Stigler's estimator (1970) of the age of the process is made.

Keywords

BRANCHING PROCESSMAXIMUM LIKELIHOOD ESTIMATIONSTIGLER'S ESTIMATORRECURRENCE FORMULA
Type
Research Papers
Information
Journal of Applied Probability , Volume 19 , Issue 4 , December 1982 , pp. 776 - 784
Copyright
Copyright © Applied Probability Trust 1982 

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

This research was supported by grants FCAC 504, NSERC A 8852 and FCAC 1608.

References

Ades, M., Dion, J. P. and Labelle, G. (1981) On estimating the age of a supercritical branching process. Technical Report, Université du Québec à Montréal.Google Scholar
Crump, K. S. and Howe, R. B. (1972) Nonparametric estimation of the age of a Galton-Watson branching process. Biometrika 59, 533–538.Google Scholar
Hwang, T. Y. and Hwang, J. T. (1978) Maximum likelihood estimate of the age of a Galton-Watson process with Poisson offspring distribution. Bull. Inst. Math. Acad. Sinica 6, 203–213.Google Scholar
Hwang, T. Y. and Wang, N. S. (1979) On best fractional linear generating function bounds. J. Appl. Prob. 16, 449–453.Google Scholar
Kojima, K. and Kelleher, T. M. (1962) Survival of mutant genes. Amer. Naturalist 96, 329–346.CrossRefGoogle Scholar
Stigler, S. M. (1970) Estimating the age of a Galton-Watson branching process. Biometrika 57, 505–512.Google Scholar