On estimating the variance of the offspring distribution in a simple branching process

1974 ◽  
Vol 6 (3) ◽  
pp. 421-433 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Rate Of Convergence ◽  
Central Limit ◽  
Branching Process ◽  
Limit Theory ◽  
Offspring Distribution ◽  
Iterated Logarithm ◽  
Single Realization ◽  
Watson Process ◽  
Limit Result ◽  
Strongly Consistent

A single realization {Zn, 0 ≦n≦N + 1} of a supercritical Galton-Watson process (so called) is considered and it is required to estimate the variance of the offspring distribution. A prospective estimator is proposed, where , and is shown to be strongly consistent on the non-extinction set. A central limit result and an iterated logarithm result are provided to give information on the rate of convergence of the estimator. It is also shown that the estimation results are robust in the sense that they continue to apply unchanged in the case where immigration occurs. Martingale limit theory is employed at each stage in obtaining the limit results.

On estimating the variance of the offspring distribution in a simple branching process

1974 ◽  
Vol 6 (03) ◽  
pp. 421-433 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Rate Of Convergence ◽  
Central Limit ◽  
Branching Process ◽  
Limit Theory ◽  
Offspring Distribution ◽  
Iterated Logarithm ◽  
Single Realization ◽  
Watson Process ◽  
Limit Result ◽  
Strongly Consistent

A single realization {Z n , 0 ≦n≦N + 1} of a supercritical Galton-Watson process (so called) is considered and it is required to estimate the variance of the offspring distribution. A prospective estimator is proposed, where , and is shown to be strongly consistent on the non-extinction set. A central limit result and an iterated logarithm result are provided to give information on the rate of convergence of the estimator. It is also shown that the estimation results are robust in the sense that they continue to apply unchanged in the case where immigration occurs. Martingale limit theory is employed at each stage in obtaining the limit results.

On the non-existence of consistent estimates in Galton-Watson processes

1982 ◽  
Vol 19 (4) ◽  
pp. 842-846 ◽  
Author(s):  
Richard Lockhart
Keyword(s):  
Offspring Distribution ◽  
Lattice Size ◽  
Single Realization ◽  
The Mean ◽  
Watson Process ◽  
Mean Variance

Estimation of the offspring distribution from a single realization of a supercritical Galton-Watson process is studied. It is shown that, based on population totals, a parameter of the offspring distribution cannot be estimated unless it is determined by the mean, variance, lattice size and lattice offset of the offspring distribution.

scholarly journals On the transient behaviour of a Poisson branching process

1967 ◽  
Vol 7 (4) ◽  
pp. 465-480 ◽  
Author(s):  
E. Seneta
Keyword(s):  
Branching Process ◽  
General Information ◽  
Transient Behaviour ◽  
Offspring Distribution ◽  
Biological Interest ◽  
Watson Process ◽  
Mean Time

SummaryIn general, information concerning the distribution of the time to absorption, T, of a simple branching (Galton-Watson) process for which extinction in finite mean time is certain, is difficult to obtain. The process of greatest biological interest is that for which the offspring distribution is Poisson, having p.g.f. F(s) = em(s-1), m < 1.

Domains of attraction for the subcritical Galton-Watson branching process

1968 ◽  
Vol 5 (01) ◽  
pp. 216-219 ◽  
Author(s):  
H. Rubin ◽  
D. Vere-Jones
Keyword(s):  
Generating Function ◽  
Branching Process ◽  
Domains Of Attraction ◽  
Offspring Distribution ◽  
Watson Process ◽  
Image Position

Let F(z) = σ fjzj be the generating function for the offspring distribution {fj } from a single ancestor in the usual Galton-Watson process. It is well-known (see Harris [1]) that if Π(z) is the generating function of the distribution of ancestors in the 0th generation, the distribution of offspring at the nth generation has generating function where F n (z), the nth functional iterate of F(z), gives the distribution of offspring at the nth generation from a single ancestor.

Some limit theorems for Markov branching processes

1973 ◽  
Vol 10 (02) ◽  
pp. 299-306 ◽  
Author(s):  
J. R. Leslie
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Original Process ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
Logarithm Law ◽  
Increasing Process ◽  
Watson Process ◽  
Iterated Logarithm Law

Analogues of the central limit theorem and iterated logarithm law have recently been obtained for the Galton-Watson process; similar results are established in this paper for the temporally homogeneous Markov branching process and for the associated increasing process consisting of the number of splits in the original process up to time t.

Domains of attraction for the subcritical Galton-Watson branching process

1968 ◽  
Vol 5 (1) ◽  
pp. 216-219 ◽  
Author(s):  
H. Rubin ◽  
D. Vere-Jones
Keyword(s):  
Generating Function ◽  
Branching Process ◽  
Domains Of Attraction ◽  
Offspring Distribution ◽  
Watson Process ◽  
Image Position

Let F(z) = σ fjzj be the generating function for the offspring distribution {fj} from a single ancestor in the usual Galton-Watson process. It is well-known (see Harris [1]) that if Π(z) is the generating function of the distribution of ancestors in the 0th generation, the distribution of offspring at the nth generation has generating function where Fn(z), the nth functional iterate of F(z), gives the distribution of offspring at the nth generation from a single ancestor.

The critical branching process with immigration stopped at zero

1985 ◽  
Vol 22 (01) ◽  
pp. 223-227 ◽  
Author(s):  
B. Gail Ivanoff ◽  
E. Seneta
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Critical Case ◽  
Initial Distribution ◽  
New Form ◽  
Watson Process ◽  
Branching Process With Immigration ◽  
Limit Result ◽  
Critical Branching Process

Limit theorems for the Galton–Watson process with immigration (BPI), where immigration is not permitted when the process is in state 0 (so that this state is absorbing), have been studied for the subcritical and supercritical cases by Seneta and Tavaré (1983). It is pointed out here that, apart from a change of context, the corresponding theorem in the critical case has been obtained by Vatutin (1977). Extensions which follow from a more general form of initial distribution are sketched, including a new form of limit result (7).

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

1982 ◽  
Vol 19 (4) ◽  
pp. 776-784 ◽  
Author(s):  
M. Adès ◽  
J.-P. Dion ◽  
G. Labelle ◽  
K. Nanthi
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Maximum Likelihood Estimate ◽  
Branching Process ◽  
Negative Binomial ◽  
Recurrence Formula ◽  
Likelihood Estimation ◽  
Likelihood Estimate ◽  
Offspring Distribution ◽  
Watson Process

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 p0 = 0 is discussed. Furthermore the maximum likelihood estimate of the age of the process when the offspring distribution is negative binomial (p0 ≠ 0) is obtained, and a comparison with Stigler's estimator (1970) of the age of the process is made.

A rate of convergence result for the super-critical Galton-Watson process

1970 ◽  
Vol 7 (2) ◽  
pp. 451-454 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Rate Of Convergence ◽  
Generating Function ◽  
Probability Generating Function ◽  
Random Variable ◽  
Convergence Result ◽  
Convergence Problem ◽  
Offspring Distribution ◽  
Watson Process ◽  
Number Of Authors

Let Z0 = 1, Z1, Z2, ··· denote a super-critical Galton-Watson process whose non-degenerate offspring distribution has probability generating function where 1 < m = EZ1 < ∞. The Galton-Watson process evolves in such a way that the generating function Fn(s) of Znis the nth functional iterate of F(s). The convergence problem for Zn, when appropriately normed, has been studied by quite a number of authors; for an ultimate form see Heyde [2]. However, no information has previously been obtained on the rate of such convergence. We shall here suppose that in which case Wn = m –nZn converges almost surely to a non-degenerate random variable W as n → ∞ (Harris [1], p. 13). It is our object to establish the following result on the rate of convergence of Wn to W.

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