Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-06T03:51:27.323Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimation theory for growth and immigration rates in a multiplicative process

Published online by Cambridge University Press:  14 July 2016

C. C. Heyde  and
E. Seneta
C. C. Heyde
Affiliation:
Australian National University
E. Seneta
Affiliation:
Australian National University
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper deals with the simple Galton-Watson process with immigration, {Xn} with offspring probability generating function (p.g.f.) F(s) and immigration p.g.f. B(s), under the basic assumption that the process is subcritical (0 < m ≡ F'(1–) < 1), and that 0 < λ ≡ B'(1–) < ∞, 0 < B(0) < 1, together with various other moment assumptions as needed. Estimation theory for the rates m and λ on the basis of a single terminated realization of the process {Xn} is developed, in that (strongly) consistent estimators for both m and λ are obtained, together with associated central limit theorems in relation to m and μ ≡ λ(1–m)–1 Following this, historical antecedents are analysed, and some examples of application of the estimation theory are discussed, with particular reference to the continuous-time branching process with immigration. The paper also contains a strong law for martingales; and discusses relation of the above theory to that of a first order autoregressive process.

Keywords

ESTIMATION THEORYGROWTH AND IMMIGRATION RATESMULTIPLICATIVE PROCESSESSTRONG CONSISTENCYSTRONG LAWMARTINGALESCENTRAL LIMIT THEOREMFIRST ORDER AUTOREGRESSIONPARTICLE FLUCTUATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 9 , Issue 2 , June 1972 , pp. 235 - 256
Copyright
Copyright © Applied Probability Trust 1972 

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.)

References

Bartlett, M. S. (1955) An Introduction to Stochastic Processes. Cambridge University Press, Cambridge.Google Scholar
Billingsley, P. (1961) Statistical Inference for Markov Processes. University of Chicago Press, Chicago.Google Scholar
Brown, B. M. (1971) Martingale central limit theorems. Ann. Math. Statist. 42, 59–66.CrossRefGoogle Scholar
Chandrasekhar, S. (1943) Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15, 1–89. (Also in Selected Papers on Noise and Stochastic Processes , ed. Wax, N., Dover, New York, 1954.)CrossRefGoogle Scholar
Fürth, R. (1918) Statistik und Wahrscheinlichkeitsnachwirkung. Z. Physik. 19, 421–426; ibid. 20 (1919) 21.Google Scholar
Hannan, E. J. (1960) Time Series Analysis. Methuen, London.Google Scholar
Harris, T. E. (1963) The Theory of Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
Heathcote, C. R. (1966) Corrections and comments on the paper “A branching process allowing immigration”. J. R. Statist. Soc. B 28, 213–217.Google Scholar
Heyde, C. C. (1971) Some central limit analogues for supercritical Galton-Watson processes. J. Appl. Prob. 8, 52–59.CrossRefGoogle Scholar
Heyde, C. C. and Seneta, E. (1971) Analogues of classical limit theorems for the supercritical Galton-Watson process with immigration. Math. Biosciences 11, 249–259.CrossRefGoogle Scholar
Loève, M. (1963) Probability Theory. 3rd Ed. Van Nostrand, New York.Google Scholar
Neveu, J. (1965) Mathematical Foundations of the Calculus of Probability. Holden-Day, San Francisco.Google Scholar
Pakes, A. G. (1971) Branching processes with immigration. J. Appl. Prob. 8, 32–42.CrossRefGoogle Scholar
Rothschild, Lord (1953) A new method of measuring the activity of spermatozoa. J. Exp. Biol. 30, 178–199.CrossRefGoogle Scholar
Seneta, E. (1969) Functional equations and the Galton-Watson process. Adv. Appl. Prob. 1, 1–42.CrossRefGoogle Scholar
Sevast'Yanov, B. A. (1957) Limit theorems for branching random processes of special form (in Russian). Teor. Veroyat. Primenen. 2, 339–348.Google Scholar
Westgren, A. (1916) Die Veränderungsgeschwindigkeit der lokalen Teilchenkonzentration in kolloiden Systemen. (Erste Mitteilung). Ark. Mat. Astronom. Fys. Band 11, No. 14, 24 pp. (Zweite Mitteilung: 1918; Band 13, No. 14, 18 pp.).Google Scholar