Asymptotic Expansions for Array Branching Processes with Applications to Bootstrapping

1998 ◽  
Vol 35 (1) ◽  
pp. 12-26 ◽  
Author(s):  
T. N. Sriram
Keyword(s):  
Distribution Function ◽  
Normal Distribution ◽  
Asymptotic Expansions ◽  
Branching Process ◽  
Branching Processes ◽  
Test Function ◽  
Branching Process With Immigration ◽  
Critical Branching Process

Asymptotic expansions are obtained for the distribution function of a studentized estimator of the offspring mean sequence in an array branching process with immigration. The expansion result is shown to hold in a test function topology. As an application of this result, it is shown that the bootstrapping distribution of the estimator of the offspring mean in a sub-critical branching process with immigration also admits the same expansion (in probability). From these considerations, it is concluded that the bootstrapping distribution provides a better approximation asymptotically than the normal distribution.

Asymptotic Expansions for Array Branching Processes with Applications to Bootstrapping

1998 ◽  
Vol 35 (01) ◽  
pp. 12-26
Author(s):  
T. N. Sriram
Keyword(s):  
Distribution Function ◽  
Normal Distribution ◽  
Asymptotic Expansions ◽  
Branching Process ◽  
Branching Processes ◽  
Test Function ◽  
Branching Process With Immigration ◽  
Critical Branching Process

Asymptotic expansions are obtained for the distribution function of a studentized estimator of the offspring mean sequence in an array branching process with immigration. The expansion result is shown to hold in a test function topology. As an application of this result, it is shown that the bootstrapping distribution of the estimator of the offspring mean in a sub-critical branching process with immigration also admits the same expansion (in probability). From these considerations, it is concluded that the bootstrapping distribution provides a better approximation asymptotically than the normal distribution.

On the convergence of the supercritical branching processes with immigration

1977 ◽  
Vol 14 (2) ◽  
pp. 387-390 ◽  
Author(s):  
Harry Cohn
Keyword(s):  
Distribution Function ◽  
Limit Distribution ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Limit Distribution Function ◽  
Branching Process With Immigration ◽  
Necessary And Sufficient ◽  
Supercritical Branching Processes

It is shown for a supercritical branching process with immigration that if the log moment of the immigration distribution is infinite, then no sequence of positive constants {cn} exists such that {Xn/cn} converges in law to a proper limit distribution function F, except for the case F(0 +) = 1. Seneta's result [1] combined with the above-mentioned one imply that if 1 < m < ∞ then the finiteness of the log moment of the immigration distribution is a necessary and sufficient condition for the existence of some constants {cn} such that {Xn/cn} converges in law to a proper limit distribution function F, with F(0 +) < 1.

scholarly journals On estimation of the variances for critical branching processes with immigration

2010 ◽  
Vol 47 (02) ◽  
pp. 526-542
Author(s):  
Chunhua Ma ◽  
Longmin Wang
Keyword(s):  
Least Squares ◽  
Branching Process ◽  
Branching Processes ◽  
Limiting Distributions ◽  
Least Squares Estimators ◽  
Conditional Least Squares ◽  
Regularly Varying ◽  
Branching Process With Immigration ◽  
Critical Branching Process

The conditional least-squares estimators of the variances are studied for a critical branching process with immigration that allows the offspring distributions to have infinite fourth moments. We derive different forms of limiting distributions for these estimators when the offspring distributions have regularly varying tails with index α. In particular, in the case in which 2 &lt; α &lt; 8/3, the normalizing factor of the estimator for the offspring variance is smaller than √n, which is different from that of Winnicki (1991).

scholarly journals Branching processes in a random environment with immigration stopped at zero

2020 ◽  
Vol 57 (1) ◽  
pp. 237-249 ◽  
Author(s):  
Elena Dyakonova ◽  
Doudou Li ◽  
Vladimir Vatutin ◽  
Mei Zhang
Keyword(s):  
Random Environment ◽  
Branching Process ◽  
Branching Processes ◽  
Time Interval ◽  
Tail Distribution ◽  
The Moment ◽  
Branching Process With Immigration ◽  
First Time ◽  
Critical Branching Process ◽  
Number Of Particles

AbstractA critical branching process with immigration which evolves in a random environment is considered. Assuming that immigration is not allowed when there are no individuals in the population, we investigate the tail distribution of the so-called life period of the process, i.e. the length of the time interval between the moment when the process is initiated by a positive number of particles and the moment when there are no individuals in the population for the first time.

On the convergence of the supercritical branching processes with immigration

1977 ◽  
Vol 14 (02) ◽  
pp. 387-390 ◽  
Author(s):  
Harry Cohn
Keyword(s):  
Distribution Function ◽  
Limit Distribution ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Limit Distribution Function ◽  
Branching Process With Immigration ◽  
Necessary And Sufficient ◽  
Supercritical Branching Processes

It is shown for a supercritical branching process with immigration that if the log moment of the immigration distribution is infinite, then no sequence of positive constants {cn } exists such that {Xn/cn } converges in law to a proper limit distribution function F, except for the case F(0 +) = 1. Seneta's result [1] combined with the above-mentioned one imply that if 1 &lt; m &lt; ∞ then the finiteness of the log moment of the immigration distribution is a necessary and sufficient condition for the existence of some constants {cn } such that {Xn /c n} converges in law to a proper limit distribution function F, with F(0 +) &lt; 1.

scholarly journals Berry-Esseen bound for nearly critical branching processes with immigration

2019 ◽  
pp. 42-49
Author(s):  
Ya. Khusanbaev ◽  
S. Sharipov ◽  
V. Golomoziy
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Rate Of Convergence ◽  
Central Limit ◽  
Branching Process ◽  
Branching Processes ◽  
Branching Process With Immigration ◽  
Critical Branching Process

In this paper, we consider a nearly critical branching process with immigration. We obtain the rate of convergence in central limit theorem for nearly critical branching processes with immigration.

scholarly journals On estimation of the variances for critical branching processes with immigration

2010 ◽  
Vol 47 (2) ◽  
pp. 526-542 ◽  
Author(s):  
Chunhua Ma ◽  
Longmin Wang
Keyword(s):  
Least Squares ◽  
Branching Process ◽  
Branching Processes ◽  
Limiting Distributions ◽  
Least Squares Estimators ◽  
Conditional Least Squares ◽  
Regularly Varying ◽  
Branching Process With Immigration ◽  
Critical Branching Process

The conditional least-squares estimators of the variances are studied for a critical branching process with immigration that allows the offspring distributions to have infinite fourth moments. We derive different forms of limiting distributions for these estimators when the offspring distributions have regularly varying tails with index α. In particular, in the case in which 2 < α < 8/3, the normalizing factor of the estimator for the offspring variance is smaller than √n, which is different from that of Winnicki (1991).

Sums of Lifetimes in Age Dependent Branching Processes

1969 ◽  
Vol 6 (01) ◽  
pp. 195-200 ◽  
Author(s):  
J. Howard Weiner
Keyword(s):  
Distribution Function ◽  
Branching Process ◽  
Past History ◽  
Branching Processes ◽  
Lifetime Distribution ◽  
Parent Cell ◽  
Absolutely Continuous ◽  
Age Dependent ◽  
A Cell ◽  
Additional Assumptions

Consider a Bellman-Harris [1] age dependent branching process. At t = 0, a cell is born, has lifetime distribution function G(t), G(0) = 0, assumed to be absolutely continuous with density g(t). At the death of the cell, k new cells are born, each proceeding independently and identically as the parent cell, and independent of past history. Denote by h(s) = Σ k=0 ∞ pk s k and suppose h(1) ≡ m, and assume h”(1) &lt; ∞. Additional assumptions will be added as required.

Supercritical age-dependent branching processes with immigration

1974 ◽  
Vol 11 (4) ◽  
pp. 695-702 ◽  
Author(s):  
K. B. Athreya ◽  
P. R. Parthasarathy ◽  
G. Sankaranarayanan
Keyword(s):  
Branching Process ◽  
Random Number ◽  
Branching Processes ◽  
Life Time ◽  
Random Variable ◽  
One Dimensional ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
Branching Process With Immigration ◽  
Mutually Independent

A branching process with immigration of the following type is considered. For every i, a random number Ni of particles join the system at time . These particles evolve according to a one-dimensional age-dependent branching process with offspring p.g.f. and life time distribution G(t). Assume . Then it is shown that Z(t) e–αt converges in distribution to an extended real-valued random variable Y where a is the Malthusian parameter. We do not require the sequences {τi} or {Ni} to be independent or identically distributed or even mutually independent.

Sums of Lifetimes in Age Dependent Branching Processes

1969 ◽  
Vol 6 (1) ◽  
pp. 195-200 ◽  
Author(s):  
J. Howard Weiner
Keyword(s):  
Distribution Function ◽  
Branching Process ◽  
Past History ◽  
Branching Processes ◽  
Lifetime Distribution ◽  
Parent Cell ◽  
Absolutely Continuous ◽  
Age Dependent ◽  
A Cell ◽  
Additional Assumptions

Consider a Bellman-Harris [1] age dependent branching process. At t = 0, a cell is born, has lifetime distribution function G(t), G(0) = 0, assumed to be absolutely continuous with density g(t). At the death of the cell, k new cells are born, each proceeding independently and identically as the parent cell, and independent of past history. Denote by h(s) = Σk=0∞pksk and suppose h(1) ≡ m, and assume h”(1) < ∞. Additional assumptions will be added as required.

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