Expectation maximization estimates of the offspring probabilities in a class of multitype branching processes with binary family trees

2017 ◽  
Vol 24 (4) ◽  
pp. 246-256
Author(s):  
Nina Daskalova
Keyword(s):  
Expectation Maximization ◽  
Branching Processes ◽  
Multitype Branching Processes ◽  
Family Trees

On multitype branching processes in a random environment

1981 ◽  
Vol 13 (3) ◽  
pp. 464-497 ◽  
Author(s):  
David Tanny
Keyword(s):  
Random Environment ◽  
Stability Condition ◽  
Branching Processes ◽  
Classification Theorem ◽  
Regularity Conditions ◽  
Exponential Rate ◽  
Probabilistic Computation ◽  
Multitype Branching Processes ◽  
The Stability ◽  
Functional Iteration

This paper is concerned with the growth of multitype branching processes in a random environment (mbpre). It is shown that, under suitable regularity conditions, the process either explodes of becomes extinct. A classification theorem is given delineating the cases of explosion or extinction. Furthermore, it is shown that the process grows at an exponential rate on its set of non-extinction provided the process is stable. Criteria is given for non-certain extinction of the mbpre to occur, and an example shows that the stability condition cannot be removed. The method of proof used, in general, is direct probabilistic computation rather than the classical functional iteration techniques. Growth theorems are first proved for increasing mbpre and subsequently transferred to general mbpre using the associated mbpre and the reduced mbpre.

scholarly journals Extinction Probabilities of Supercritical Decomposable Branching Processes

2012 ◽  
Vol 49 (03) ◽  
pp. 639-651 ◽  
Author(s):  
Sophie Hautphenne
Keyword(s):  
Nonnegative Solution ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Equivalence Classes ◽  
Specific Class ◽  
Initial Particle ◽  
Whole Process ◽  
Extinction Probabilities ◽  
The Minimal Nonnegative Solution ◽  
Multitype Branching Processes

We focus on supercritical decomposable (reducible) multitype branching processes. Types are partitioned into irreducible equivalence classes. In this context, extinction of some classes is possible without the whole process becoming extinct. We derive criteria for the almost-sure extinction of the whole process, as well as of a specific class, conditionally given the class of the initial particle. We give sufficient conditions under which the extinction of a class implies the extinction of another class or of the whole process. Finally, we show that the extinction probability of a specific class is the minimal nonnegative solution of the usual extinction equation but with added constraints.

scholarly journals Extinction Probabilities of Supercritical Decomposable Branching Processes

2012 ◽  
Vol 49 (3) ◽  
pp. 639-651 ◽  
Author(s):  
Sophie Hautphenne
Keyword(s):  
Nonnegative Solution ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Equivalence Classes ◽  
Specific Class ◽  
Initial Particle ◽  
Whole Process ◽  
Extinction Probabilities ◽  
The Minimal Nonnegative Solution ◽  
Multitype Branching Processes

We focus on supercritical decomposable (reducible) multitype branching processes. Types are partitioned into irreducible equivalence classes. In this context, extinction of some classes is possible without the whole process becoming extinct. We derive criteria for the almost-sure extinction of the whole process, as well as of a specific class, conditionally given the class of the initial particle. We give sufficient conditions under which the extinction of a class implies the extinction of another class or of the whole process. Finally, we show that the extinction probability of a specific class is the minimal nonnegative solution of the usual extinction equation but with added constraints.

Asymptotic behavior of near-critical multitype branching processes

1991 ◽  
Vol 28 (03) ◽  
pp. 512-519 ◽  
Author(s):  
Fima C. Klebaner
Keyword(s):  
Asymptotic Behavior ◽  
Population Size ◽  
Discrete Time ◽  
Growth Rates ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Limiting Distribution ◽  
Size Dependent ◽  
Generalized Gamma ◽  
Multitype Branching Processes

Sufficient conditions for survival and extinction of multitype population-size-dependent branching processes in discrete time are obtained. Growth rates are determined on the set of divergence to infinity. The limiting distribution of a properly normalized process can be generalized gamma, normal or degenerate.

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