Statistical inference of 2-type critical Galton–Watson processes with immigration

Körmendi, Kristóf; Pap, Gyula

doi:10.1007/s11203-016-9148-y

Cited by 2 publications

(5 citation statements)

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“…By developing a better understanding of the ideas related to this decomposition we managed to tackle the general case, where we only assume that the ospring mean matrix is positively regular. These results can be found in Körmendi and Pap [16,Theorem 3.1] and are also reproduced in Subsection 4.2. We nish this section with a new result: We examine the asymptotic properties of a joint estimator of both the ospring mean matrix and the immigration mean.…”

Section: Introductionsupporting

confidence: 80%

“…So when the non-degeneracy condition fails, then both ospring distributions are degenerate. In this thesis we prove results under (ND), however we note, that Körmendi and Pap [16] contains some results under the degenerate case as well.…”

Section: A Limit Theorem For the Processmentioning

confidence: 76%

“…It was proved using our toolkit, however in a more obscure form, not as streamlined and clearly structured as it is presented here. Later we treated the general case [16], those results are reproduced in Subsection 4.2. Finally Subsection 4.3 oers a small extension in treating the immigration mean as unknown parameter.…”

Section: A Discussion Of the Estimatesmentioning

confidence: 91%

“…We introduce the random variables U k in (12) as the wellEehved part of our decomposition, we know their limiting behaviour, it coincides with the underlying 1-dimensional stochastic process in Theorem 2.4. Then we introduce the random variables V k in (16) as the prolemti part of the decomposition.This part doesn't contribute to the limit of the process, but as we will see it later it does in case of the estimator.…”

Section: A Summarymentioning

confidence: 99%

“…Quine and Durham [21] described a strongly consistent and asymptotically normal estimator for the ospring mean matrix in the subcritical case, while Shete and Sriram [22] established similar results in the supercritical case, however in the supercritical case the estimator requires more information than just the generation sizes of each type of individual. Results in the critical case was rst established by Ispány et al [10] under heavy restrictions on the structure of the ospring mean matrix, then later Körmendi and Pap [16] lifted the restrictions and described the asymptotic behaviour of an estimator in both the critical and subcritical cases.…”

Section: Introductionmentioning

confidence: 99%

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Estimation of the offspring means for critical 2-type Galton-Watson processes with immigration

Körmendi

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Section: Introductionsupporting

confidence: 80%

Section: A Limit Theorem For the Processmentioning

confidence: 76%

Section: A Discussion Of the Estimatesmentioning

confidence: 91%

Section: A Summarymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Estimation of the offspring means for critical 2-type Galton-Watson processes with immigration

Körmendi

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Asymptotic behavior of critical indecomposable multi-type branching processes with immigration

Danka

Pap

2016

ESAIM: PS

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In this paper the asymptotic behavior of a critical multi-type branching process with immigration is described when the offspring mean matrix is irreducible, in other words, when the process is indecomposable. It is proved that sequences of appropriately scaled random step functions formed from periodic subsequences of a critical indecomposable multi-type branching process with immigration converge weakly towards a process supported by a ray determined by the Perron vector of the offspring mean matrix. The types can be partitioned into nonempty mutually disjoint subsets (according to communication of types) such that the coordinate processes belonging to the same subset are multiples of the same squared Bessel process, and the coordinate processes belonging to different subsets are independent.where X (n) t := n −1 X ⌊nt⌋ for t ∈ R + , n ∈ N, and (X t ) t∈R + is the pathwise unique strong solution of the stochastic differential equation (SDE)with initial value X 0 = 0, where m ε := E(ε 1 ), V ξ := Var(ξ 1,1 ), (W t ) t∈R + is a standard Wiener process, and x + denotes the positive part of x ∈ R.A multi-type branching process (X k ) k∈Z + is referred to respectively as subcritical, critical or supercritical if ̺(m ξ ) < 1, ̺(m ξ ) = 1 or ̺(m ξ ) > 1, where ̺(m ξ ) denotes the spectral radius of the offspring mean matrix m ξ (see, e.g., Athreya and Ney [1] or Quine [22]). Joffe and Métivier [15, Theorem 4.3.1] studied a critical multi-type branching process without immigration when the offspring mean matrix is primitive, in other words, when the process is positively regular. They determined the limiting behavior of the martingale part (M (n) ) n∈N given by M (n) t := n −1 ⌊nt⌋ k=1 M k with M k := X k −E(X k | X 0 , . . . , X k−1 ) (which is a special case of Theorem 3.2).The result (1.2) has been generalized by Ispány and Pap [11] for a critical p-type branching process (X k ) k∈Z + with immigration when the offspring mean matrix is primitive, in other words, when the process is positively regular. They proved that

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Statistical inference of 2-type critical Galton–Watson processes with immigration

Cited by 2 publications

References 26 publications

Estimation of the offspring means for critical 2-type Galton-Watson processes with immigration

Estimation of the offspring means for critical 2-type Galton-Watson processes with immigration

Asymptotic behavior of critical indecomposable multi-type branching processes with immigration

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