Estimation of the parameters of a branching process from migrating binomial observations

1998 ◽  
Vol 30 (4) ◽  
pp. 948-967 ◽  
Author(s):  
C. Jacob ◽  
J. Peccoud
Keyword(s):  
Confidence Interval ◽  
Asymptotic Behaviour ◽  
Population Size ◽  
Branching Process ◽  
Total Population ◽  
Initial Population ◽  
Asymptotic Confidence Interval ◽  
Total Population Size ◽  
Initial Population Size ◽  
Watson Process

This paper considers a branching process generated by an offspring distribution F with mean m < ∞ and variance σ2 < ∞ and such that, at each generation n, there is an observed δ-migration, according to a binomial law Bpvn*Nnbef which depends on the total population size Nnbef. The δ-migration is defined as an emigration, an immigration or a null migration, depending on the value of δ, which is assumed constant throughout the different generations. The process with δ-migration is a generation-dependent Galton-Watson process, whereas the observed process is not in general a martingale. Under the assumption that the process with δ-migration is supercritical, we generalize for the observed migrating process the results relative to the Galton-Watson supercritical case that concern the asymptotic behaviour of the process and the estimation of m and σ2, as n → ∞. Moreover, an asymptotic confidence interval of the initial population size is given.

Estimation of the parameters of a branching process from migrating binomial observations

1998 ◽  
Vol 30 (04) ◽  
pp. 948-967 ◽  
Author(s):  
C. Jacob ◽  
J. Peccoud
Keyword(s):  
Confidence Interval ◽  
Asymptotic Behaviour ◽  
Population Size ◽  
Branching Process ◽  
Total Population ◽  
Initial Population ◽  
Asymptotic Confidence Interval ◽  
Total Population Size ◽  
Initial Population Size ◽  
Watson Process

This paper considers a branching process generated by an offspring distribution F with mean m &lt; ∞ and variance σ2 &lt; ∞ and such that, at each generation n, there is an observed δ-migration, according to a binomial law B p v n *N n bef which depends on the total population size N n bef. The δ-migration is defined as an emigration, an immigration or a null migration, depending on the value of δ, which is assumed constant throughout the different generations. The process with δ-migration is a generation-dependent Galton-Watson process, whereas the observed process is not in general a martingale. Under the assumption that the process with δ-migration is supercritical, we generalize for the observed migrating process the results relative to the Galton-Watson supercritical case that concern the asymptotic behaviour of the process and the estimation of m and σ2, as n → ∞. Moreover, an asymptotic confidence interval of the initial population size is given.

scholarly journals Wright–Fisher-like models with constant population size on average

2017 ◽  
Vol 10 (06) ◽  
pp. 1750078 ◽  
Author(s):  
Nicolas Grosjean ◽  
Thierry Huillet
Keyword(s):  
Population Genetics ◽  
Population Size ◽  
Discrete Time ◽  
Total Population ◽  
Neutral Evolution ◽  
Bottleneck Effect ◽  
Total Population Size ◽  
Constant Size ◽  
Basic Facts ◽  
Watson Process

We first recall some basic facts from the theory of discrete-time Markov chains arising from two types neutral and non-neutral evolution models of population genetics with constant size. We then define and analyze a version of such models whose fluctuating total population size is conserved on average only. In our model, the population of interest is seen as being embedded in a frame process which is a critical Galton–Watson process. In this context, we address problems such as extinction, fixation, size of the population at fixation and survival probability to a bottleneck effect of the environment.

Limit theorems for a general critical branching process

1971 ◽  
Vol 8 (01) ◽  
pp. 1-16 ◽  
Author(s):  
Stephen D. Durham
Keyword(s):  
Life Span ◽  
Population Size ◽  
Limit Theorems ◽  
Branching Process ◽  
Total Population ◽  
Life Time ◽  
Multiple Births ◽  
Total Population Size ◽  
Initial Object ◽  
Critical Branching Process

A general branching process begins with an initial object born at time 0. The initial object lives a random length of time and, during its life-time, has offspring which reproduce and die as independent probabilistic copies of the parent. Number and times of births to a parent are random and, once an object is born, its behavior is assumed to be independent of all other objects, independent of total population size and independent of absolute time. The life span of a parent and the number and times its offspring arrive may be interdependent. Multiple births are allowed. The process continues as long as there are objects alive.

The age of a Galton-Watson population with a geometric offspring distribution

2002 ◽  
Vol 39 (4) ◽  
pp. 816-828 ◽  
Author(s):  
F. C. Klebaner ◽  
S. Sagitov
Keyword(s):  
Population Size ◽  
Asymptotic Distribution ◽  
Initial Population ◽  
Asymptotic Results ◽  
The Past ◽  
Offspring Distribution ◽  
Initial Population Size ◽  
Watson Process ◽  
Looking Back

Motivated by the question of the age in a branching population we try to recreate the past by looking back from the currently observed population size. We define a new backward Galton-Watson process and study the case of the geometric offspring distribution with parameter p in detail. The backward process is then the Galton-Watson process with immigration, again with a geometric offspring distribution but with parameter 1-p, and it is also the dual to the original Galton-Watson process. We give the asymptotic distribution of the age when the initial population size is large in supercritical and critical cases. To this end, we give new asymptotic results on the Galton-Watson immigration processes stopped at zero.

Asymptotic results for the extinction time of Markov branching processes allowing emigration, I. Random walk decrements

1989 ◽  
Vol 21 (02) ◽  
pp. 243-269 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Population Size ◽  
Large Scale ◽  
Branching Process ◽  
Branching Processes ◽  
Initial Population ◽  
Extinction Time ◽  
Asymptotic Results ◽  
Time To Extinction ◽  
Initial Population Size ◽  
The Mathematical Model

The mathematical model is a Markov branching process which is subjected to catastrophes or large-scale emigration. Catastrophes reduce the population size by independent and identically distributed decrements, and two mechanisms for generating catastrophe epochs are given separate consideration. These are that catastrophes occur at a rate proportional to population size, and as an independent Poisson process. The paper studies some properties of the time to extinction of the modified process in those cases where extinction occurs almost surely. Particular attention is given to limit theorems and the behaviour of the expected extinction time as the initial population size grows. These properties are contrasted with known properties for the case when there is no catastrophe component.

Asymptotic growth of a class of size-and-age-dependent birth processes

1974 ◽  
Vol 11 (2) ◽  
pp. 248-254 ◽  
Author(s):  
W. A. O'N. Waugh
Keyword(s):  
Population Size ◽  
Branching Process ◽  
Total Population ◽  
Asymptotic Expression ◽  
Branching Processes ◽  
Total Population Size ◽  
Stochastic Population Models ◽  
Age Dependent ◽  
Birth Processes ◽  
Special Case

A class of binary fission stochastic population models is described, in which the fission probabilities may depend on the age of an individual and the total population size. Age-dependent binary branching processes with Erlangian lifelength distributions are a special case. An asymptotic expression for the growth of the population size is developed, which generalizes known theorems about the asymptotic exponential growth of a branching process.

Asymptotic results for the extinction time of Markov branching processes allowing emigration, I. Random walk decrements

1989 ◽  
Vol 21 (2) ◽  
pp. 243-269 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Population Size ◽  
Large Scale ◽  
Branching Process ◽  
Branching Processes ◽  
Initial Population ◽  
Extinction Time ◽  
Asymptotic Results ◽  
Time To Extinction ◽  
Initial Population Size ◽  
The Mathematical Model

The mathematical model is a Markov branching process which is subjected to catastrophes or large-scale emigration. Catastrophes reduce the population size by independent and identically distributed decrements, and two mechanisms for generating catastrophe epochs are given separate consideration. These are that catastrophes occur at a rate proportional to population size, and as an independent Poisson process.The paper studies some properties of the time to extinction of the modified process in those cases where extinction occurs almost surely. Particular attention is given to limit theorems and the behaviour of the expected extinction time as the initial population size grows. These properties are contrasted with known properties for the case when there is no catastrophe component.

Limit theorems for a general critical branching process

1971 ◽  
Vol 8 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Stephen D. Durham
Keyword(s):  
Life Span ◽  
Population Size ◽  
Limit Theorems ◽  
Branching Process ◽  
Total Population ◽  
Life Time ◽  
Multiple Births ◽  
Total Population Size ◽  
Initial Object ◽  
Critical Branching Process

A general branching process begins with an initial object born at time 0. The initial object lives a random length of time and, during its life-time, has offspring which reproduce and die as independent probabilistic copies of the parent. Number and times of births to a parent are random and, once an object is born, its behavior is assumed to be independent of all other objects, independent of total population size and independent of absolute time. The life span of a parent and the number and times its offspring arrive may be interdependent. Multiple births are allowed. The process continues as long as there are objects alive.

The age of a Galton-Watson population with a geometric offspring distribution

2002 ◽  
Vol 39 (04) ◽  
pp. 816-828 ◽  
Author(s):  
F. C. Klebaner ◽  
S. Sagitov
Keyword(s):  
Population Size ◽  
Asymptotic Distribution ◽  
Initial Population ◽  
Asymptotic Results ◽  
The Past ◽  
Offspring Distribution ◽  
Initial Population Size ◽  
Watson Process ◽  
Looking Back

Motivated by the question of the age in a branching population we try to recreate the past by looking back from the currently observed population size. We define a new backward Galton-Watson process and study the case of the geometric offspring distribution with parameter p in detail. The backward process is then the Galton-Watson process with immigration, again with a geometric offspring distribution but with parameter 1-p, and it is also the dual to the original Galton-Watson process. We give the asymptotic distribution of the age when the initial population size is large in supercritical and critical cases. To this end, we give new asymptotic results on the Galton-Watson immigration processes stopped at zero.

Asymptotic growth of a class of size-and-age-dependent birth processes

1974 ◽  
Vol 11 (02) ◽  
pp. 248-254 ◽  
Author(s):  
W. A. O'N. Waugh
Keyword(s):  
Population Size ◽  
Branching Process ◽  
Total Population ◽  
Asymptotic Expression ◽  
Branching Processes ◽  
Total Population Size ◽  
Stochastic Population Models ◽  
Age Dependent ◽  
Birth Processes ◽  
Special Case

A class of binary fission stochastic population models is described, in which the fission probabilities may depend on the age of an individual and the total population size. Age-dependent binary branching processes with Erlangian lifelength distributions are a special case. An asymptotic expression for the growth of the population size is developed, which generalizes known theorems about the asymptotic exponential growth of a branching process.

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