Estimation for multitype branching processes

1977 ◽  
Vol 14 (04) ◽  
pp. 829-835 ◽  
Author(s):  
M. P. Quine ◽  
P. Durham
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Natural Conditions ◽  
Multitype Branching Process ◽  
Large Samples ◽  
The Central Limit Theorem ◽  
The Matrix ◽  
Multitype Branching Processes ◽  
Branching Process With Immigration ◽  
Strongly Consistent

It is shown that certain estimators of the matrix of offspring means and the vector of stationary means in a subcritical multitype branching process with immigration are strongly consistent and obey the central limit theorem under fairly natural conditions. The results give some evidence of the robustness of the analogous autoregressive time series model in the case of large samples.

Estimation for multitype branching processes

1977 ◽  
Vol 14 (4) ◽  
pp. 829-835 ◽  
Author(s):  
M. P. Quine ◽  
P. Durham
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Natural Conditions ◽  
Multitype Branching Process ◽  
Large Samples ◽  
The Central Limit Theorem ◽  
The Matrix ◽  
Multitype Branching Processes ◽  
Branching Process With Immigration ◽  
Strongly Consistent

It is shown that certain estimators of the matrix of offspring means and the vector of stationary means in a subcritical multitype branching process with immigration are strongly consistent and obey the central limit theorem under fairly natural conditions. The results give some evidence of the robustness of the analogous autoregressive time series model in the case of large samples.

Conditions for extinction in certain bisexual Galton–Watson branching processes

1984 ◽  
Vol 21 (02) ◽  
pp. 414-418
Author(s):  
David M. Hull
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Multitype Branching Process ◽  
Extinction Probabilities ◽  
Multitype Branching Processes ◽  
Necessary And Sufficient

A multitype branching process, the n-family community mating process, is introduced for the purpose of comparing extinction probabilities with those of bisexual Galton–Watson branching processes. Consideration of known properties of standard multitype branching processes leads to conditions which are both necessary and sufficient for extinction in a bisexual Galton–Watson branching process. An application is then made to the counterexample of the author's earlier paper.

scholarly journals An Algorithmic Approach to Modelling the Co-Evolution of Parasites and Their Hosts

2020 ◽  
Vol 9 (2) ◽  
pp. 13
Author(s):  
Charles J. Mode
Keyword(s):  
Host Plants ◽  
Branching Process ◽  
Branching Processes ◽  
Focus Of Attention ◽  
Point Of View ◽  
Simulation Experiments ◽  
Multitype Branching Process ◽  
Algorithmic Approach ◽  
Multitype Branching Processes ◽  
Growing Crop

This paper is a reformulation of the paper, Mode 1958 Evolution 12:158 - 165, which was written in terms of a deterministic paradigm, using di erential equations In this paper, however, the working paradigm will be stochastic, and from the mathematical point of view, it will be a stochastic process that may be viewed as a branching process within a branching process. In particular, it will be assumed that the population of host plants will evolve as a multitype branching process, and the pathogen, which grows on the leaves of the host in every generation of the host, will also be assumed to evolve as a multitype branching processes during each generation of the host. The contents of this paper, were motivated by problems in Agriculture in which Plant Pathologists and Plant Breeders work together to control the damage inflicted by a pathogen on a growing crop of a cultivar such as flax, wheat. and many other cultivars. The focus of attention in this paper is the development of algorithms that will guide the development of software to run Monte Carol simulation experiments taking into account mutations in the host and pathogen. The writing of software to implement the algorithms developed in this paper would require a major e ort, and is, therefore, beyond the scope of this paper

Conditions for extinction in certain bisexual Galton–Watson branching processes

1984 ◽  
Vol 21 (2) ◽  
pp. 414-418 ◽  
Author(s):  
David M. Hull
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Multitype Branching Process ◽  
Extinction Probabilities ◽  
Multitype Branching Processes ◽  
Necessary And Sufficient

A multitype branching process, the n-family community mating process, is introduced for the purpose of comparing extinction probabilities with those of bisexual Galton–Watson branching processes. Consideration of known properties of standard multitype branching processes leads to conditions which are both necessary and sufficient for extinction in a bisexual Galton–Watson branching process. An application is then made to the counterexample of the author's earlier paper.

Large deviations for supercritical multitype branching processes

2004 ◽  
Vol 41 (03) ◽  
pp. 703-720 ◽  
Author(s):  
Owen Dafydd Jones
Keyword(s):  
Large Deviations ◽  
Growth Rates ◽  
Branching Process ◽  
Large Deviation ◽  
Branching Processes ◽  
Log P ◽  
Single Individual ◽  
Periodic Functions ◽  
Multitype Branching Process ◽  
Multitype Branching Processes

Large deviation results are obtained for the normed limit of a supercritical multitype branching process. Starting from a single individual of type i, let L[i] be the normed limit of the branching process, and let be the minimum possible population size at generation k. If is bounded in k (bounded minimum growth), then we show that P(L[i] ≤ x) = P(L[i] = 0) + x α F *[i](x) + o(x α ) as x → 0. If grows exponentially in k (exponential minimum growth), then we show that −log P(L[i] ≤ x) = x −β/(1−β) G*[i](x) + o (x −β/(1−β)) as x → 0. If the maximum family size is bounded, then −log P(L[i] > x) = x δ/(δ−1) H *[i](x) + o(x δ/(δ−1)) as x → ∞. Here α, β and δ are constants obtained from combinations of the minimum, maximum and mean growth rates, and F *, G * and H * are multiplicatively periodic functions.

Multitype branching processes based on exact progeny lengths of particles in a Galton-Watson branching process

1989 ◽  
Vol 26 (01) ◽  
pp. 1-8
Author(s):  
V. G. Gadag ◽  
M. B. Rajarshi
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
White Male ◽  
Asymptotic Results ◽  
Male Population ◽  
Multitype Branching Process ◽  
Original Process ◽  
The Usa ◽  
Multitype Branching Processes ◽  
Lines Of Descent

In Gadag and Rajarshi (1987), we studied a bivariate (multitype) branching process based on infinite and finite lines of descent, of particles of a supercritical one-dimensional (multitype) Galton-Watson branching process (GWBP). In this paper, we discuss a few more meaningful and interesting univariate and multitype branching processes, based on exact progeny lengths of particles in a GWBP. Our constructions relax the assumption of supercriticality made in Gadag and Rajarshi (1987). We investigate some finite-time and asymptotic results of these processes in some details and relate them to the original process. These results are then used to propose new and better estimates of the offspring mean. An illustration based on the branching process of the white male population of the USA is also given. We believe that our work offers a rather finer understanding of the branching property.

Multitype branching processes based on exact progeny lengths of particles in a Galton-Watson branching process

1989 ◽  
Vol 26 (1) ◽  
pp. 1-8 ◽  
Author(s):  
V. G. Gadag ◽  
M. B. Rajarshi
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
White Male ◽  
Asymptotic Results ◽  
Male Population ◽  
Multitype Branching Process ◽  
Original Process ◽  
The Usa ◽  
Multitype Branching Processes ◽  
Lines Of Descent

In Gadag and Rajarshi (1987), we studied a bivariate (multitype) branching process based on infinite and finite lines of descent, of particles of a supercritical one-dimensional (multitype) Galton-Watson branching process (GWBP). In this paper, we discuss a few more meaningful and interesting univariate and multitype branching processes, based on exact progeny lengths of particles in a GWBP. Our constructions relax the assumption of supercriticality made in Gadag and Rajarshi (1987). We investigate some finite-time and asymptotic results of these processes in some details and relate them to the original process. These results are then used to propose new and better estimates of the offspring mean. An illustration based on the branching process of the white male population of the USA is also given. We believe that our work offers a rather finer understanding of the branching property.

scholarly journals Multitype branching processes observing particles of a given type

2005 ◽  
Vol 42 (2) ◽  
pp. 446-462 ◽  
Author(s):  
Claudia Ceci ◽  
Anna Gerardi
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Markov Property ◽  
Suitable Assumption ◽  
Recursive Structure ◽  
Multitype Branching Process ◽  
Weak Uniqueness ◽  
Multitype Branching Processes ◽  
Branching Property ◽  
Number Of Particles

A multitype branching process is presented in the framework of marked trees and its structure is studied by applying the strong branching property. In particular, the Markov property and the expression for the generator are derived for the process whose components are the numbers of particles of each type. The filtering of the whole population, observing the number of particles of a given type, is discussed. Weak uniqueness for the filtering equation and a recursive structure for the linearized filtering equation are proved under a suitable assumption on the reproduction law.

Large deviations for supercritical multitype branching processes

2004 ◽  
Vol 41 (3) ◽  
pp. 703-720 ◽  
Author(s):  
Owen Dafydd Jones
Keyword(s):  
Large Deviations ◽  
Growth Rates ◽  
Branching Process ◽  
Large Deviation ◽  
Branching Processes ◽  
Log P ◽  
Single Individual ◽  
Periodic Functions ◽  
Multitype Branching Process ◽  
Multitype Branching Processes

Large deviation results are obtained for the normed limit of a supercritical multitype branching process. Starting from a single individual of type i, let L[i] be the normed limit of the branching process, and let be the minimum possible population size at generation k. If is bounded in k (bounded minimum growth), then we show that P(L[i] ≤ x) = P(L[i] = 0) + xαF*[i](x) + o(xα) as x → 0. If grows exponentially in k (exponential minimum growth), then we show that −log P(L[i] ≤ x) = x−β/(1−β) G*[i](x) + o (x−β/(1−β)) as x → 0. If the maximum family size is bounded, then −log P(L[i] > x) = xδ/(δ−1)H*[i](x) + o(xδ/(δ−1)) as x → ∞. Here α, β and δ are constants obtained from combinations of the minimum, maximum and mean growth rates, and F*, G* and H* are multiplicatively periodic functions.

scholarly journals Multitype branching processes observing particles of a given type

2005 ◽  
Vol 42 (02) ◽  
pp. 446-462
Author(s):  
Claudia Ceci ◽  
Anna Gerardi
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Markov Property ◽  
Suitable Assumption ◽  
Recursive Structure ◽  
Multitype Branching Process ◽  
Weak Uniqueness ◽  
Multitype Branching Processes ◽  
Branching Property ◽  
Number Of Particles

A multitype branching process is presented in the framework of marked trees and its structure is studied by applying the strong branching property. In particular, the Markov property and the expression for the generator are derived for the process whose components are the numbers of particles of each type. The filtering of the whole population, observing the number of particles of a given type, is discussed. Weak uniqueness for the filtering equation and a recursive structure for the linearized filtering equation are proved under a suitable assumption on the reproduction law.

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